\begin{Theorem} \label{thm:shuffspe} Assume that $\delta \in \mathcal{S}_m$ and  $\pi \in \mathcal{S}_n$  are two disjoint   permutations, where $\des(\delta) = r$ and $\des(\pi) = s$. Moreover, $\delta_1<\delta_2$ and all of the elements of $\delta$ are larger than the elements of $\pi$. Then
\begin{align*}
  (1)&   \sum_{\substack{\alpha \in \pi \shuffle_l \delta  \\ \mathrm{des}(\alpha) = d}} q^{\mathrm{maj}(\alpha)}= {m-r+s \brack d-r} {n-s+r-1\brack d-s-1} \times q^{\mathrm{maj}(\delta) + \mathrm{maj}(\pi)  + (d - s)(d - r)}, \\[5pt]
    (2)&  \sum_{\substack{\alpha \in \pi \shuffle_{ls} \delta  \\ \mathrm{des}(\alpha) = d}} q^{\mathrm{maj}(\alpha)}= {m-r+s-1 \brack d-r} {n-s+r-1\brack d-s-1} \times q^{\mathrm{maj}(\delta) + \mathrm{maj}(\pi)  + (d - s)(d - r)}, \\[5pt]
   (3) &   \sum_{\substack{\alpha \in \pi \shuffle_{ll} \delta  \\ \mathrm{des}(\alpha) = d}} q^{\mathrm{maj}(\alpha)}= {m-r+s-1 \brack d-r} {n-s+r-1\brack d-s-1} \times q^{\mathrm{maj}(\delta) + \mathrm{maj}(\pi)  + (d- s+1)(d - r)}.
\end{align*}
\end{Theorem}

\begin{Theorem}\label{th:main}
The trivariate statistic {\rm (}$\ides$, $\des$, $\maj${\rm )}  are equidistributed over  the set of $n$-Andr\'e I permutations,  $n$-Andr\'e II permutations and $n$-simsun permutations, i.e.,
   \begin{align*}
\sum_{\sigma \in \AndI_n} s^{\ides(\sigma)}t^{\des(\sigma)}  q^{\maj(\sigma)}
	&=\sum_{\sigma \in \AndII_n} s^{\ides(\sigma)}t^{\des(\sigma)}  q^{\maj(\sigma)}\\[5pt]
    &=\sum_{\sigma \in \RS_{n}} s^{\ides(\sigma)}t^{\des(\sigma)} q^{\maj(\sigma)}.
\end{align*}

\end{Theorem}

In the field of enumerative combinatorics, several kinds of permutations are counted by {\it Euler numbers}, such as {\it alternating permutations}, {\it Andr\'e I and II permutations}, and {\it simsun  permutations}. {\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\sec(x)+\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\'e} in the late 19th century \cite{andre1881permutations}. Andr\'e proved that $E_n$ counts the number of {\it alternating permutations} of length $n$ (see \cite{stanley2009survey}), which are permutations $\sigma = \sigma_1\sigma_2\ldots\sigma_n$ satisfying $\sigma_1>\sigma_2<\sigma_3>\cdots$.

\medskip

Andr\'e permutations were first introduced by Foata and Sch\"utzenberger and further studied by Strehl \cite{Str74} and Foata and Strehl \cite{FSt74, FSt76}.   For clarity, we will work with permutations of length $n$ for which each permutation is a sequence of $n$ distinct integers not necessarily from 1 to $n$.  The empty word $e$ and any single-letter word are defined as both {\it Andr\'e I permutations} and {\it Andr\'e II permutations}.
For a permutation $\sigma=\sigma_1\sigma_2\cdots \sigma_n$ ($n\geq 2$) of length $n$, we decompose it as $\sigma=\tau\,\min(\sigma)\,\tau'$. Here $\sigma$ is the concatenation of a left factor~$\tau$, followed by the minimum letter  $\min(\sigma)$, and a right factor $\tau'$.  Then, $\sigma$ is called an {\it Andr\'e I permutation} (resp. {\it Andr\'e II permutation}) if both $\tau$ and $\tau'$ are   Andr\'e I permutations (resp. Andr\'e II permutations), and the maximum letter of the subword $\tau\tau'$ lies in $\tau'$ (resp. the minimum letter of $\tau\tau'$ lies in $\tau'$).

\medskip

The set of all Andr\'e I permutations on the set $[n]:=\{1,2,\ldots, n\}$ is denoted by $\AndI_{n}$ and the set of  Andr\'e II  permutations  on the set $[n]$ is denoted by  $\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\'e I permutations and Andr\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\AndI_n| = |\AndII_n|$.

\smallskip
Andr\'e I permutations for $n\leq 5$ are listed below:

$n=1$:\quad 1;\qquad $n=2$:\quad 12;\qquad
$n=3$:\quad 123, 213;

$n=4$:\quad 1234, 1324, 2314, 2134, 3124;

$n=5$:\quad 12345, 12435, 13425, 23415, 13245, 14235, 34125,
24135,\hfil\break
\indent\hphantom{$n=5$:\quad}23145, 21345, 41235, 31245, 21435, 32415, 41325, 31425.

\smallskip
Andr\'e II permutations for $n\leq 5$ are listed below:

$n=1$:\quad 1;\qquad $n=2$:\quad 12;\qquad
$n=3$:\quad 123, 312;

$n=4$:\quad 1234, 1423, 3412, 4123, 3124;

$n=5$:\quad 12345, 12534, 14523, 34512, 15234, 14235, 34125,
45123,\hfil\break
\indent\hphantom{$n=5$:\quad}35124, 51234, 41235, 31245, 51423, 53412, 41523, 31524.

\medskip

 Simsun  permutations were introduced by
 Rodica Simion and Sheila Sundaram in a series of studies of homology representations of the symmetric group \cite{sundaram1995homology,sundaram1996plethysm}. To better elaborate on our results,  we adopt the following definition of simsun  permutations.  A permutation $\sigma = \sigma_1\sigma_2\ldots\sigma_n$ on the set $[n]$ is called a simsun  permutation if $\sigma_n=n$ and it contains no double descents, and this property is preserved after removing the elements $n, n-1, n-2, \ldots, 1$ in order.
For example, it is easy to see that $\sigma= 21473658$ is a simsun  permutation since $21473658$, $2147365$, $214365$, $21435$, $2143$, $213$, $21$, $1$ have no double descents. Recall that  an index $i$ (where $1 \le i < n$)  is called a {\it descent} of a permutation $\sigma=\sigma_1\ldots \sigma_n$ if $\sigma_i > \sigma_{i+1}$ and an  index $i$ (where $1 \le i \leq n-2$)  is called a {\it double descent} if $\sigma_i > \sigma_{i+1}>\sigma_{i+2}$.

 Notably, if one removes the last element from a simsun  permutation  as defined here, the resulting permutation aligns with the original definition of simsun  permutations due to Simion and Sundaram.

The set of all simsun  permutations on the set $[n]$ is denoted by $\RS_{n}$.  A remarkable property of simsun  permutations is that  $|\RS_n|=E_{n}$. The notation $\RS_{n}$ was first adopted by Chow
and Shiu  \cite{chow2011counting}.

Simsun  permutations for $n\leq 5$ are listed below:

$n=1$:\quad 1;\qquad $n=2$:\quad 12; \qquad $n=3$:\quad 123, 213;

$n=4$:\quad 1234, 1324, 2134, 2314, 3124;

 $n=5$:\quad 12345, 12435, 13245, 13425, 14235, 21345, 21435, 23145,\hfil\break
\indent\hphantom{$n=5$:\quad}23415, 24135, 31245, 31425, 34125, 41235, 41325, 42315.

\medskip

Andr\'e permutations and simsun  permutations provide  new combinatorial interpretations for the Euler numbers.
They  play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}.
Other properties about Andr\'e permutations and simsun  permutations have been extensively studied by Barnabei et al. \cite{barnabei2020permutations}, Chow and Shiu \cite{chow2011counting}, Deutsch-Elizalde  \cite{deutsch2012restricted},  Disanto \cite{Di14},  Foata and the first author  \cite{FH01} and so on.     In particular, by constructing a bijection between the set of  Andr\'e I permutations and the set of simsun  permutations,  Chow and Shiu \cite{chow2011counting} observed that the number of descents are equidistributed over  Andr\'e I permutations and  simsun  permutations. Specifically, let $\des(\sigma)$ denote the number of descents of $\sigma$, they showed that
$$
	 \sum_{\sigma \in \AndI_n}   t^{\des(\sigma)} =\sum_{\sigma \in \RS_{n}}  t^{\des(\sigma)}.
$$

In this paper, we show that the number of inverse descents are also equidistributed over  Andr\'e permutations and  simsun  permutations.  The number of inverse descents of a permutation $\sigma$ is simply the number of descents of its inverse permutation $\sigma^{-1}$, namely, $
\mathrm{ides}(\sigma) = \mathrm{des}(\sigma^{-1}).$ In fact,  we show that the trivariate statistic ($\ides$, $\des$, $\maj$)  are equidistributed over  Andr\'e permutations and  simsun  permutations, where  {\it  the major index} $\maj(\sigma)$ of $\sigma$ is defined  to be the sum of its descents of $\sigma$.    For brevity,   we adopt the notation
   $n$-Andr\'e permutations for Andr\'e permutations  on   $[n]$ and $n$-simsun  permutations for   simsun permutations on   $[n]$.
\medskip

Our main result is as follows.

In the field of enumerative combinatorics, several kinds of permutations are counted by {\it Euler numbers}, such as {\it alternating permutations}, {\it Andr\'e I and II permutations}, and {\it simsun permutations}. {\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\sec(x)+\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\'e} in the late 19th century \cite{andre1881permutations}. Andr\'e proved that $E_n$ counts the number of {\it alternating permutations} of length $n$ (see \cite{stanley2009survey}), which are permutations $\sigma = \sigma_1\sigma_2\ldots\sigma_n$ satisfying $\sigma_1>\sigma_2<\sigma_3>\cdots$.

The set of all Andr\'e I permutations on the set $[n]:=\{1,2,\ldots, n\}$ is denoted by $\AndI_{n}$ and the set of Andr\'e II permutations on the set $[n]$ is denoted by $\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\'e I permutations and Andr\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\AndI_n| = |\AndII_n|$.

The set of all simsun permutations on the set $[n]$ is denoted by $\RS_{n}$. A remarkable property of simsun permutations is that $|\RS_n|=E_{n}$. The notation $\RS_{n}$ was first adopted by Chow
and Shiu \cite{chow2011counting}.

Andr\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers.
They play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}.
Other properties about Andr\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \cite{barnabei2020permutations}, Chow and Shiu \cite{chow2011counting}, Deutsch-Elizalde \cite{deutsch2012restricted}, Disanto \cite{Di14}, Foata and the first author \cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\'e I permutations and the set of simsun permutations, Chow and Shiu \cite{chow2011counting} observed that the number of descents are equidistributed over Andr\'e I permutations and simsun permutations. Specifically, let $\des(\sigma)$ denote the number of descents of $\sigma$, they showed that
$$
 \sum_{\sigma \in \AndI_n} t^{\des(\sigma)} =\sum_{\sigma \in \RS_{n}} t^{\des(\sigma)}.
$$

In this paper, we show that the number of inverse descents are also equidistributed over Andr\'e permutations and simsun permutations. The number of inverse descents of a permutation $\sigma$ is simply the number of descents of its inverse permutation $\sigma^{-1}$, namely, $
\mathrm{ides}(\sigma) = \mathrm{des}(\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\ides$, $\des$, $\maj$) are equidistributed over Andr\'e permutations and simsun permutations, where {\it the major index} $\maj(\sigma)$ of $\sigma$ is defined to be the sum of its descents of $\sigma$. For brevity, we adopt the notation
 $n$-Andr\'e permutations for Andr\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.
\medskip

Our main result is as follows.

\begin{document}
 \begin{abstract}
Simsun permutations, Andr\'e I permutations and Andr\'e II permutations are three combinatorial models for Euler numbers. It's known that
the descent statistic is equidistributed
over the set of Andr\'e I permutations and the set of simsun permutations. In this paper, we prove that
the trivariate statistic $(\ides, \des, \maj)$, comprising the inverse descent, descent, and major index,
are equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of Andr\'e I permutations and the set of Andr\'e II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem.
 \end{abstract}
 \maketitle

Andr\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers.
They play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}.
Other properties about Andr\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \cite{barnabei2020permutations}, Chow and Shiu \cite{chow2011counting}, Deutsch-Elizalde \cite{deutsch2012restricted}, Disanto \cite{Di14}, Foata and the first author \cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\'e I permutations and the set of simsun permutations, Chow and Shiu \cite{chow2011counting} observed that the number of descents are equidistributed over Andr\'e I permutations and simsun permutations. Specifically, let $\des(\sigma)$ denote the number of descents of $\sigma$, they showed that
$$
 \sum_{\sigma \in \AndI_n} t^{\des(\sigma)} =\sum_{\sigma \in \RS_{n}} t^{\des(\sigma)}.
$$

In this paper, we show that the number of inverse descents are also equidistributed over Andr\'e permutations and simsun permutations. The number of inverse descents of a permutation $\sigma$ is simply the number of descents of its inverse permutation $\sigma^{-1}$, namely, $
\mathrm{ides}(\sigma) = \mathrm{des}(\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\ides$, $\des$, $\maj$) are equidistributed over Andr\'e permutations and simsun permutations, where {\it the major index} $\maj(\sigma)$ of $\sigma$ is defined to be the sum of its descents of $\sigma$. For brevity, we adopt the notation
 $n$-Andr\'e permutations for Andr\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.
\medskip

Our main result is as follows.

\medskip

To our knowledge, even the special case of the above result for the univariate statistic ``$\ides$" is new:
$$
 A_n(s):= \sum_{\sigma \in \AndI_n} s^{\ides(\sigma)}
 =\sum_{\sigma \in \AndII_n} s^{\ides(\sigma)}
 =\sum_{\sigma \in \RS_{n}} s^{\ides(\sigma)}.
$$
We list the first values of the polynomials $A_n(s)$ below:
\begin{align*}
A_1(s)&=1, \quad A_2(s)=1,\quad A_3(s)=s + 1,\quad
A_4(s)=4s + 1,\\
A_5(s)&=4s^2 + 11s + 1,\quad
A_6(s)=2s^3 + 32s^2 + 26s + 1.
\end{align*}

\begin{Theorem}\label{th:equi-shape} For $n\geq 1$ and any unlabeled binary tree $T \in \URL_n$, we have
 \begin{align}
 \sum_{\sigma \in \AndI(T)} s^{\ides(\sigma)} \overset{(a)}{=}\sum_{\sigma \in \AndII(T)} s^{\ides(\sigma)} \overset{(b)}{=}\sum_{\sigma \in \RS(T)} s^{\ides(\sigma)}.
\end{align}
\end{Theorem}

\noindent{\it Proof of relation {\rm (}a{\rm )} in Theorem~ \ref{th:equi-shape}.} We proceed by induction on $n$. For $n=1$, relation (a) clearly holds. Assume that it holds for all $p<n$. We aim to show that it also holds for $n$. Let $T$ be an unlabeled (rooted) binary tree in $\URL_n$ with the left subtree $T^l$ and the right subtree $T^r$ of the root, respectively.
By the definition of Andr\'e permutations, we have
\[\AndI(T)=\bigcup_{\sigma \in \AndI(T^l)} \bigcup_{ \tau \in \AndI(T^r)} \sigma \vartriangle \tau \]
and
\[\AndII(T)=\bigcup_{\hat{\sigma} \in \AndII(T^l)} \bigcup_{ \hat{\tau} \in \AndII(T^l)} \hat{\sigma} \triangledown \hat{\tau}. \]
This implies that
\begin{align}
\sum_{\mu \in \AndI(T)}t^{\ides(\mu)}&=\sum_{\sigma \in \AndI(T^l) } \sum_{\tau \in \AndI(T^r)}\sum_{\mu \in \sigma \vartriangle \tau}t^{\ides(\mu)} \label{pf:lem7m1} \\[5pt]
\sum_{\mu \in \AndII(T)}t^{\ides(\mu)}&=\sum_{\hat{\sigma} \in \AndII(T^l) } \sum_{\hat{\tau} \in \AndII(T^r)}\sum_{\mu \in \hat{\sigma}\triangledown \hat{\tau}}t^{\ides(\mu)}. \label{pf:lem7m2}
\end{align}
Note that ${T}^l$ and $T^r$ are the left and right subtrees of the root of $T$, so their vertices are less than $n$.
The induction hypothesis
implies that there exists a bijection $\phi^l$ between $\AndI({T}^l)$ and $\AndII({T}^l)$ such that for $\sigma \in \AndI({T}^l)$ and $\phi(\sigma) \in \AndII({T}^l)$, we have $\ides(\sigma)=\ides(\phi^l(\sigma)).$
Similarly, there exists a bijection $\phi^r$ between $\AndI({T}^r)$ and $\AndII({T}^r)$ such that for $\tau \in \AndI({T}^r)$ and $\phi^r(\tau ) \in \AndII({T}^r)$, we have $\ides(\tau)=\ides(\phi^r(\tau )).$
Hence by Corollary~\ref{cor:ideseq:2}, we arrive at \begin{equation} \label{pf:lem7m3}
\sum_{\mu \in \sigma \vartriangle \tau}t^{\ides(\mu )} =\sum_{\mu \in \phi^l(\sigma) \triangledown \phi^r(\tau)}t^{\ides(\mu)}.
\end{equation}
We therefore derive that
\begin{align*}
\sum_{\mu \in \AndI(T)}t^{\ides(\mu)}&\stackrel{\eqref{pf:lem7m1}}{=}\sum_{\sigma \in \AndI(T^l) } \sum_{\tau \in \AndI(T^r)}\sum_{\mu \in \sigma \vartriangle \tau}t^{\ides(\mu)} \\[5pt]
&\stackrel{\eqref{pf:lem7m3}}{=}\sum_{\phi^l(\sigma) \in \AndII(T^l) } \sum_{\phi^r(\tau) \in \AndII(T^r)}\sum_{\mu \in \phi^l(\sigma) \triangledown \phi^r(\tau)}t^{\ides(\mu)} \\[5pt]
&\stackrel{\eqref{pf:lem7m2}}{=}\sum_{\mu \in \AndII(T)}t^{\ides(\mu)}.
\end{align*}
This confirms that relation (a) also holds for $n$. Thus, we complete the proof of Theorem~ \ref{th:equi-shape} (a). \qed

\begin{Theorem}\label{th:equi-shape} For $n\geq 1$ and any  unlabeled binary tree $T \in \URL_n$, we have
  \begin{align}
 	 \sum_{\sigma \in \AndI(T)}   s^{\ides(\sigma)} \overset{(a)}{=}\sum_{\sigma \in \AndII(T)}  s^{\ides(\sigma)} \overset{(b)}{=}\sum_{\sigma \in \RS(T)}  s^{\ides(\sigma)}.
\end{align}
\end{Theorem}

\medskip

To our knowledge, even the special case of the above result  for the univariate statistic ``$\ides$" is new:
$$
	A_n(s):=	\sum_{\sigma \in \AndI_n} s^{\ides(\sigma)}
	=\sum_{\sigma \in \AndII_n} s^{\ides(\sigma)}
    =\sum_{\sigma \in \RS_{n}} s^{\ides(\sigma)}.
$$
We list the first values of the polynomials $A_n(s)$ below:
\begin{align*}
A_1(s)&=1, \quad A_2(s)=1,\quad A_3(s)=s + 1,\quad
A_4(s)=4s + 1,\\
A_5(s)&=4s^2 + 11s + 1,\quad
A_6(s)=2s^3 + 32s^2 + 26s + 1.
\end{align*}

\medskip

The proof of Theorem \ref{th:main} can be sketched as follows: By   sending Andr\'e permutations and simsun  permutations to increasing binary trees and applying Proposition \ref{th:equi-shape-des} in Section 2,  the proof of Theorem \ref{th:main}  reduces to showing that the  inverse descents (ides) is equidistributed  over the $n$-Andr\'e I permutations, the $n$-Andr\'e II permutations
and the $n$-simsun  permutations that share the same tree shape (see Theorem \ref{th:equi-shape} in Section 2). The proof of Theorem \ref{th:equi-shape} is split  into two parts: (a) proving the equidistribution of $\ides$  over
 $n$-Andr\'e I  permutations and  $n$-Andr\'e II  permutations with the same tree shape (relation  (a) in Theorem \ref{th:equi-shape}),  and (b)   proving the equidistribution of $\ides$  over
the  $n$-Andr\'e II  permutations and the simsun  permutations with the same tree shape  (relation  (b) in Theorem \ref{th:equi-shape}). Specially, the proof of relation  (a)   relies on an investigation of    the shuffle of permutations  (see Section 3 for details), while the proof of  relation (b) proceeds by   constructing a bijection between the set of  $n$-Andr\'e II permutations and the set of the $n$-simsun  permutations (see Section 4). It would be interesting to give a direct explicit bijective proof of relation  (a).

\medskip

In the proof of relation  (a) in Theorem \ref{th:equi-shape}, the following general result plays an important role. Note that this result applies to ordinary permutations, not solely Andr\'e permutations.

Let $\mathfrak{S}_n$ denote the set of permutations on the set $[n]$. Suppose that $\sigma \in \mathfrak{S}_j$ and $\tau \in \mathfrak{S}_k$. We define the following three sets of permutations:
\begin{align*}
\sigma \lozenge\tau &=\{ \mu = \sigma'1\tau' \in \mathfrak{S}_{j+k+1} \mid
\sigma' \sim \sigma,\ \tau' \sim \tau \};\\
\sigma \vartriangle \tau
&=\{ \mu = \sigma'1\tau' \in \mathfrak{S}_{j+k+1} \mid
\sigma' \sim \sigma,\ \tau' \sim \tau,\ j+k+1\in \tau' \};\\
\sigma \triangledown \tau
&=\{ \mu = \sigma'1\tau' \in \mathfrak{S}_{j+k+1} \mid
\sigma' \sim \sigma,\ \tau' \sim \tau,\ 2\in \tau' \},
\end{align*}
where $\sigma' \sim \sigma$ means that reducing the letters of  $\sigma'$ to $\{1,2,\ldots,j\}$ yields $\sigma$.

For example, if $\mu=\sigma'1\tau' =692581473$, then
$\sigma'=69258$, which reduces to $\sigma=35124$, $\tau'=473$ which reduces to  $\tau=231$.

\begin{Theorem}\label{thm:ideseq:3}
Let  $\sigma \in \mathfrak{S}_j$ and $\tau\in \mathfrak{S}_k$
be two permutations with
$\ides(\sigma)=j'$ and $\ides(\tau)=k'$. Then
\begin{align} \label{eqn:ideseq:pfc}
\sum_{\mu \in \sigma \lozenge \tau} t^{\ides(\mu)}&=\sum_{i\geq 1} \binom{k-k'+j'+1}{i+j'-k'} \binom{j-j'+k'-1}{i-1} t^{i+j'},\\[5pt]
\sum_{\mu \in \sigma \vartriangle \tau} t^{\ides(\mu)}&=\sum_{i\geq 1} \binom{k-k'+j'}{i+j'-k'} \binom{j-j'+k'-1}{i-1} t^{i+j'}, \label{eqn:ideseq:pfccc} \\[5pt]
\sum_{\mu \in \sigma \triangledown \tau} t^{\ides(\mu)}&=\sum_{i\geq 1} \binom{k-k'+j'}{i+j'-k'} \binom{j-j'+k'-1}{i-1} t^{i+j'}. \label{eqn:ideseq:pfcc}
\end{align}

\end{Theorem}

Remark. When $t=1$, we derive  the following two identities, which also follow from the Chu-Vandermonde identity \cite[Example 1.1.17]{Stanley-EC1-2012}:
\begin{align*}
\binom{j+k}{j} &=
\sum_{i\geq 1} \binom{j-d-1}{i-1}
\binom{ k+d+1}{k-i+1} ,\\
\binom{j+k-1}{j} &=\sum_{i\geq 1}\binom{j-d-1}{i-1}
\binom{ k+d}{k-i} .
\end{align*}

By considering the inverses of permutations, Theorem \ref{thm:ideseq:3} can be transformed to special cases of three refinements of Stanley's shuffle theorem (see  Theorem
\ref{thm:shuffspe}). Stanley's shuffle theorem was first established by Stanley \cite{Stanley-1972} in his study of $P$-partitions. As observed by  Gessel and Zhuang \cite{Gessel-Zhuang-2018}, this theorem
implies that  the major index (maj) and descent number (des)  are shuffle compatible, which has motivated several recent works, including those by  Adin, Gessel, Reiner  and Roichman \cite{Adin-Gessel-Reiner-Roichman-2021}, Baker-Jarvis and Sagan \cite{Baker-Sagan-2020}, Domagalski, Liang, Minnich and Sagan \cite{Domagalski-Liang-Minnich-2021}, Grinberg \cite{Grinberg-2018},  the second author and Zhang \cite{Ji-Zhang-2022} and Yang and Yan \cite{Yang-Yan-2022}.
Bijective proofs of Stanley's  Shuffle Theorem have been given by  Goulden \cite{Goulden-1985}, the second author and Zhang \cite{Ji-Zhang-2024} and Stadler \cite{Stadler-1999}.  In particular,  the second author and Zhang \cite{Ji-Zhang-2024} established several refinements of this theorem based on their bijections. The proof of Theorem \ref{thm:ideseq:3} in this paper also relies on their bijection, see Section 4 for more details.

\bigskip

The proof of Theorem \ref{th:main} is sketched as follows. First, by “sending Andr\'e permutations and simsun permutations to increasing binary trees” and applying Proposition \ref{th:equi-shape-des} (Section 2), the proof “reduces to showing that the inverse descents (ides) is equidistributed over the $n$-Andr\'e I permutations, the $n$-Andr\'e II permutations and the $n$-simsun permutations that share the same tree shape” (Theorem \ref{th:equi-shape}). The proof of Theorem \ref{th:equi-shape} is then split into two parts: (a) equidistribution of $\ides$ for $n$-Andr\'e I vs. $n$-Andr\'e II with the same tree shape (relation (a)), and (b) equidistribution of $\ides$ for $n$-Andr\'e II vs. $n$-simsun with the same tree shape (relation (b)). For (a), the sketch says the proof “relies on an investigation of the shuffle of permutations” (Section 3); for (b), it “proceeds by constructing a bijection between the set of $n$-Andr\'e II permutations and the set of the $n$-simsun permutations” (Section 4).

To prove the main theorem, we proceed as follows. First, by “sending Andr\'e permutations and simsun permutations to increasing binary trees” and applying the following proposition.

\begin{Proposition} \label{th:equi-shape-des} Let $\sigma$ be a permutation, and let $T_\sigma=\Psi(\sigma)$ be the increasing binary tree corresponding to $\sigma$ under the bijection $\Psi$. Then the descent set $\Des(\sigma)$ of $\sigma$ is completely determined by  the shape of $T_\sigma$.
\end{Proposition}

Using this (as indicated in Section 2), the proof “reduces to showing that the inverse descents (ides) is equidistributed over the $n$-Andr\'e I permutations, the $n$-Andr\'e II permutations and the $n$-simsun permutations that share the same tree shape”, i.e. to proving the following theorem.

\begin{Theorem}\label{th:equi-shape} For $n\geq 1$ and any  unlabeled binary tree $T \in \URL_n$, we have
  \begin{align}
 	 \sum_{\sigma \in \AndI(T)}   s^{\ides(\sigma)} \overset{(a)}{=}\sum_{\sigma \in \AndII(T)}  s^{\ides(\sigma)} \overset{(b)}{=}\sum_{\sigma \in \RS(T)}  s^{\ides(\sigma)}.
\end{align}
\end{Theorem}

The proof of this theorem is then split into two parts: (a) equidistribution of $\ides$ for $n$-Andr\'e I vs. $n$-Andr\'e II with the same tree shape (relation (a)), and (b) equidistribution of $\ides$ for $n$-Andr\'e II vs. $n$-simsun with the same tree shape (relation (b)). For (a), the sketch says the proof “relies on an investigation of the shuffle of permutations” (as indicated in Section 3); for (b), it “proceeds by constructing a bijection between the set of $n$-Andr\'e II permutations and the set of the $n$-simsun permutations” (as indicated in Section 4). This completes the proof of the main theorem.

Question: For every positive integer \(n\), let \([n]=\{1,2,\dots,n\}\), and let \(\AndI_n\), \(\AndII_n\), and \(\RS_n\) denote respectively the sets of André I permutations, André II permutations, and simsun permutations of \([n]\). For a permutation \(\sigma=\sigma_1\sigma_2\cdots\sigma_n\), let \(\des(\sigma)\) be the number of descents \(i\) such that \(\sigma_i>\sigma_{i+1}\), let \(\ides(\sigma)=\des(\sigma^{-1})\), and let \(\maj(\sigma)\) be the sum of all descent positions of \(\sigma\). Which statement holds for every \(n\)?

The joint distribution of \((\ides,\des,\maj)\) is the same on \(\AndI_n\), \(\AndII_n\), and \(\RS_n\); equivalently,
\[
\sum_{\sigma\in\AndI_n}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\AndII_n}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\RS_n}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)}.
\]

Choices:
B.
The joint distribution of \((\ides,\des,\maj)\) is the same on \(\AndI_n\) and \(\AndII_n\), and after forgetting \(\ides\) it is also the same on \(\RS_n\); equivalently,
\[
\sum_{\sigma\in\AndI_n}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\AndII_n}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)},
\qquad
\sum_{\sigma\in\AndI_n}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\RS_n}t^{\des(\sigma)}q^{\maj(\sigma)}.
\]
C.
The bivariate distribution of \((\des,\maj)\) is the same on \(\AndI_n\), \(\AndII_n\), and \(\RS_n\); equivalently,
\[
\sum_{\sigma\in\AndI_n}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\AndII_n}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\RS_n}t^{\des(\sigma)}q^{\maj(\sigma)}.
\]
D.
For every unlabeled binary tree \(T\in\URL_n\), the joint distribution of \((\ides,\des,\maj)\) is the same on the shape classes \(\AndI(T)\), \(\AndII(T)\), and \(\RS(T)\); equivalently,
\[
\sum_{\sigma\in\AndI(T)}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\AndII(T)}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)}
=
\sum_{\sigma\in\RS(T)}s^{\ides(\sigma)}t^{\des(\sigma)}q^{\maj(\sigma)}.
\]
E.
There exists a bijection \(\phi_n:\AndI_n\to\AndII_n\to\RS_n\) preserving each of \(\ides\), \(\des\), and \(\maj\) individually; in particular,
\[
\ides(\sigma)=\ides(\phi_n(\sigma)),\qquad \des(\sigma)=\des(\phi_n(\sigma)),\qquad \maj(\sigma)=\maj(\phi_n(\sigma))
\]
for all \(\sigma\), and hence the three generating functions in \(s,t,q\) are equal.

shape-based reduction only gives full conclusion after proving ides equidistribution on simsun as well

dropped the inverse-descent variable \(s^{\ides}\)

per-shape theorem concerns only \(\ides\), not full \((\ides,\des,\maj)\) within each fixed tree shape

equidistribution proved via shape reduction, shuffle refinements, and a bijection only in one part, not by a single global statistic-preserving bijection

The stem defines the statistics and classes but does not reveal the theorem-level conclusion. It asks the test-taker to distinguish among several nearby claims, so there is no explicit or trivial answer leakage.

This is not a bare restatement of a single theorem in the stem. The options include weaker, stronger, and differently scoped variants, so the task is to identify the precise valid conclusion rather than echo a stated fact.

The item creates strong generative pressure: the choices differ in subtle ways involving joint vs marginal distribution, global vs shape-wise statements, and equidistribution vs statistic-preserving bijection. Selecting the correct option requires real discrimination and theorem-level reasoning.

The distractors are mathematically plausible and target common failure modes: accepting a weaker true statement, overextending to shape classes, or confusing equidistribution with existence of a bijection preserving each statistic. They are distinct and well-aligned with likely misconceptions.

High-quality MCQ. It avoids answer leakage, is non-tautological, and uses strong, nuanced distractors that genuinely test precise mathematical understanding.


Let $F=\left( F_{1}, \ldots , F_{m}\right)$ be a random vector whose components are differentiable in the Malliavin sense. Denote by $ \Gamma_{F}$ its Malliavin matrix, whose entries are given by
 $\Gamma_{F}(i,j)= \langle DF_{i}, DF_{j}\rangle _{\HHH}$, $1\leq i, j\leq m$,
where $D$ denotes the Malliavin derivative with respect to an isonormal process $(W(h), h\in \HHH)$ on a real and separable Hilbert space $\HHH$. A classical result in Malliavin calculus (the {\it Bouleau-Hirsch criterion}) states that if $\Gamma_{F}$ is almost surely invertible, then $F$ admits a density with respect to the Lebesgue measure on $\mathbb{R} ^{m}$.

The random vector $F$ is said to be nondegenerate if its components are infinitely Malliavin differentiable and if $\left( \det \Gamma_{F}\right) ^{-1}\in L ^{p}(\Omega)$ for every $p\geq 1$. In this case, its density is smooth and belongs to the Schwartz space of rapidly decreasing $C ^{\infty}$ functions on $ \mathbb{R} ^{m}$. Hence, obtaining a smooth density is tightly linked to the existence of negative moments of the Malliavin derivatives of the components.

When the components of $F$ lie in a fixed Wiener chaos, the above result can be strengthened. It was shown in \cite{NNP} that in this setting the almost sure invertibility of $\Gamma_F$ is actually equivalent to the absolute continuity of the law of $F$. Moreover, \cite{NT} proves that a pair of multiple stochastic integrals fails to admit a density if and only if its components are proportional, mirroring the Gaussian case.

In the recent work \cite{LNNT} we applied the Bouleau-Hirsch criterion to establish the absolute continuity of finite-dimensional distributions of Hermite processes. Let $q\geq 1$ be an integer and $ (Z ^{H, q}_{t}, t\in \mathbb{R})$ a Hermite process of order $q$ with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Recall that $Z^{H,q}$ is an $H$-self-similar process with stationary increments and long memory, living in the $q$th Wiener chaos. The class of Hermite processes includes fractional Brownian motion ($q=1$), which is Gaussian, and the Rosenblatt process ($q=2$), which belongs to the second Wiener chaos. For any $m\geq 1$ and any grid $0<t_{1},...<t _{m}$, \cite{LNNT} shows that the determinant of the Malliavin matrix of the Hermite vector $Z= \left( Z^{H, q}_{t_{1}}, \ldots , Z^{H, q}_{t_{m}}\right)$ is almost surely strictly positive; therefore, $Z$ admits a density.

The objective of the present work is to go a step further and investigate the smoothness of this density. As previously mentioned, smoothness is deeply linked to nondegeneracy, and in particular to the existence of negative moments of the determinant of the Malliavin matrix. While establishing these negative moments appears challenging for general $q$, we succeed in proving nondegeneracy for $q=2$, that is for Rosenblatt vectors. This relies crucially on the specific structure of random variables in the second Wiener chaos, which can be represented as weighted sums of independent centered chi-square random variables.

Our first main result concerns the smoothness of the density of increments of the Rosenblatt process.

Our first main result concerns the smoothness of the density of increments of the Rosenblatt process.

As a consequence,
for all $m\geq 1$ and for all $0\leq s_{1}<...< s_{m}$, the vector $(Z_{s_{1}},\ldots ,Z_{s_{m}})$ also has a density in $ \mathcal{S}(\mathbb{R} ^{m})$, see Corollary \ref{cor1}. The proof of Theorem \ref{tt3} relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.

\begin{theorem}\label{tt2}
 Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. For any $m\geq 1$ and $n=(n_{1},..., n_{m})\in \mathbb{N}^{m}$, there exists $C>0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}<t_{1}<...<t_{m}$, the density $p_{\pi} $ of the vector $Z_{\pi}$ defined in (\ref{zpi}) satisfies
 \begin{equation*}
 \left| \partial _{x} ^{n} p_{\pi }(x)\right| \leq C \prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-H(1+ n_{j})} e ^{-c\frac{x_{j}}{(t_{j}- t_{j-1}) ^{H}}},
 \end{equation*}
 with $c=c(m,q)>0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$
\end{theorem}

\begin{theorem}
 Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 2$ and $0=t_{0}<t_{1}<\dots<t_{m}$. Consider the random vector
 \begin{equation*}
 Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, Z_{t_{2}}-Z_{t_{1}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right).
 \end{equation*}
Then
\begin{equation*}
 P\left( \det \Gamma _{Z_{\pi} }>0\right)= 1.
\end{equation*}
In particular, the random vector $Z_{\pi}$ admits a density with respect to the Lebesgue measure on $\mathbb{R} ^{m}$.
\end{theorem}

\begin{lemma}\label{ll2}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 2$ and $0=t_{0}<t_{1}<\dots<t_{m}$. For $j\geq 1$, set
\begin{equation}
\label{21o-1}
\tilde{E}_{j-1}= \span \{ D(Z_{t_{k}}-Z_{t_{l}}): k,l=0,1,\dots, j-1\}.
\end{equation}
Then, for every $j \geq 1$,
\begin{equation*}
\left\| D(Z_{t_{j}}-Z_{t_{j-1}})- {\rm proj}_{\tilde{E}_{j-1}} \big(D(Z_{t_{j}}-Z_{t_{j-1}})\big)\right\| _{\HHH} ^{2} \overset{\rm law}{=} (t_{j}-t_{j-1})^{2H} F_{j},
\end{equation*}
where, for each $j\geq 1$, $F_{j}$ is a random variable such that $F_{j} \geq \Vert DZ_{1}\Vert ^{2}_{\HHH_{1}}$, with $\HHH_{1}= L ^{2}([0, 1])$.
\end{lemma}

\begin{lemma}\label{ll3}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0=t_{0}<t_{1}<\dots<t_{m}$. Consider the random vector
\begin{equation*}
Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
\end{equation*}
and let $\Gamma_{Z_{\pi}}$ be the Malliavin matrix of $Z _{\pi }$. Then, for any integer $k\geq 0$ and any $p\geq 1$,
\begin{equation}\label{kpnorm:invdet}
 \left\| \left( \det \Gamma_{Z_{\pi}}\right) ^{-1} \right\| _{k, p} \leq C \prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-2H}.
\end{equation}
\end{lemma}

\begin{corollary}\label{cor1}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0<s_{1}<\dots<s_{m}$. Then the vector $(Z_{s_{1}},\dots, Z_{s_{m}})$ admits a density which belongs to the Schwartz space $\mathcal{S}(\mathbb{R}^{m})$.
\end{corollary}

\begin{proposition}\label{pp2}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $n, m\geq 1$ be integers and fix $0=t_{0}<t_{1}<\dots<t_{m}$ and $\beta = \left( \beta_{1},\dots, \beta_{n} \right) \in \{1,\dots, m\}^{n}$. Let $Z_{\pi}$ be given by \eqref{zpi}. Then there exists a random variable $H ^{\beta} \in \mathbb{D} ^{\infty}$ such that, for every $\varphi \in C_{p} ^{\infty}(\mathbb{R} ^{m})$,
\begin{equation}\label{22o-1}
\mathbf{E} \left[ \partial _{\beta } \varphi (Z_{\pi})\right] = \mathbf{E}\left[ \varphi (Z_{\pi})H ^{\beta}\right].
\end{equation}
Moreover,
\begin{equation}\label{22o-2}
\mathbf{E}\left[ (H ^{\beta}) ^{2} \right]\leq C \prod_{j=1}^{n}( t_{\beta_{j}}-t_{\beta _{j-1}})^{-2H}.
\end{equation}
\end{proposition}

\begin{theorem}\label{tt3}
	Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector
	\begin{equation}\label{zpi}
		Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
	\end{equation}
	Then $Z_{\pi }$ admits a density belonging to the Schwartz space  $ \mathcal{S}(\mathbb{R} ^{m})$ of rapidly decreasing smooth functions.
\end{theorem}

\begin{equation}\label{zpi}
		Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
	\end{equation}

As a consequence,
for all  $m\geq 1$ and for all $0\leq s_{1}<...< s_{m}$, the vector $(Z_{s_{1}},\ldots ,Z_{s_{m}})$ also has a density in $ \mathcal{S}(\mathbb{R} ^{m})$, see Corollary \ref{cor1}. The proof of Theorem \ref{tt3} relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.

Our second main result provides exponential-type bounds for the partial derivatives of the density  of the  finite-dimensional distributions of the Rosenblatt process. For $m \in \mathbb{N}^{\ast}$  and  $n=(n_{1},..., n_{m})\in \mathbb{N} ^{m}$, set
\begin{equation*}
	\partial _{x} ^{n} =\prod_{j=1}^{m} \left( \frac{\partial}{\partial x_{j}}\right) ^{n_{j}}.
\end{equation*}

\begin{theorem}\label{tt2}
	Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$.  For any  $m\geq 1$ and $n=(n_{1},..., n_{m})\in \mathbb{N}^{m}$, there exists $C>0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}<t_{1}<...<t_{m}$, the density $p_{\pi} $ of the vector $Z_{\pi}$  defined in (\ref{zpi})  satisfies
	\begin{equation*}
		\left| \partial _{x} ^{n} p_{\pi }(x)\right| \leq C \prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-H(1+ n_{j})} e ^{-c\frac{x_{j}}{(t_{j}- t_{j-1}) ^{H}}},
	\end{equation*}
	with $c=c(m,q)>0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$
\end{theorem}

The proof of
Theorem \ref{tt2} is based on a Malliavin integration by parts formula expressing the partial derivatives of $p_{\pi}$ (see relation (\ref{22o-1}) in Proposition \ref{pp2} below)  and on a precise control of the resulting terms using Malliavin calculus and tail estimates for random variables in Wiener chaos.

The paper is organized  as follows. Section 2 provides the necessary preliminaries on Malliavin calculus, Wiener chaos and the Rosenblatt process. Section 3 contains the proof of the smoothness of the density of Rosenblatt vectors. Section 4 is devoted to the analysis of the regularity of the partial derivatives of this density.

The proof of Theorem \ref{tt3} "relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space."

The proof of the main theorem relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.

\label{tt3}
	Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector
	\begin{equation}\label{zpi}
		Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
	\end{equation}
	Then $Z_{\pi }$ admits a density belonging to the Schwartz space  $ \mathcal{S}(\mathbb{R} ^{m})$ of rapidly decreasing smooth functions.,

Question:
Let $(Z_t, t\in\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in\left(\tfrac12,1\right)$. Let $m\ge 1$ and choose times $0=t_0<t_1<\cdots<t_m$. Define the increment vector
\[
Z_{\pi}=\bigl(Z_{t_1}-Z_{t_0},\; Z_{t_2}-Z_{t_1},\; \ldots,\; Z_{t_m}-Z_{t_{m-1}}\bigr)\in\mathbb{R}^m.
\]
Which statement holds for every such Rosenblatt process and every such grid? Here $\mathcal S(\mathbb R^m)$ denotes the Schwartz space of $C^\infty$ functions on $\mathbb R^m$ whose derivatives decay rapidly at infinity.

The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$, and that density belongs to $\mathcal S(\mathbb R^m)$.

Choices:
B. The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$, and this density is $C^{\infty}$ on $\mathbb R^m$ with all partial derivatives bounded by constants depending only on $m$ and $H$.
C. The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$.
D. If the Malliavin matrix $\Gamma_{Z_{\pi}}$ is almost surely invertible, then $Z_{\pi}$ admits a density belonging to $\mathcal S(\mathbb R^m)$.
E. The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$, and its density belongs to $\mathcal S(\mathbb R^m)$ provided $m\ge 2$.

The stem does not reveal the correct option explicitly or by a trivial cue. Defining the increment vector and Schwartz space supplies needed context, not the answer itself.

The correct choice is very close to a theorem-style statement about Rosenblatt-process increments, so the item largely asks for recall/recognition of the exact conclusion rather than a substantially reformulated consequence.

Some reasoning is needed to distinguish the strongest valid statement from a weaker true statement and from subtly overstrengthened false variants, but the item mainly tests theorem recognition rather than deeper generative inference.

The distractors are plausible and mathematically targeted: one is weaker-but-true, others alter regularity, add an incorrect sufficient-condition upgrade, or introduce a false boundary restriction. These reflect common failure modes.

A solid MCQ with no real answer leakage and strong distractors, but it is fairly close to a direct theorem restatement and only moderately probes generative reasoning.


For $q\geq 0$, the poset     $\W_n^q$ is a lattice for any $n\geq 1$.

Let $q$ be a non-negative real number. A $q$-{\it decreasing word} of length $n\geq 0$ is a binary word of length $n$ satisfying the following constraint:

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows  that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have  $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
        $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
    with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors  of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

The notion of $q$-decreasing words has recently attracted significant attention in the literature. This family of words exhibits a striking combinatorial property whenever $q$ is a positive integer. Indeed, they are in one-to-one correspondence with binary strings that avoid the pattern $1^{q+1}$, i.e. binary strings without $q+1$ consecutive 1 (see \cite{bkv gray code}). So, this implies that $q$-decreasing words of length $n$ (when $q$ is a positive integer) are enumerated by the $(q+1)$-generalized Fibonacci numbers $F_{n+1}^{q+1}$ where $F_n^q$ is defined by

$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\sum_{n\geq 0}F_n^qx^n=\frac{1}{1-x-x^2-\cdots -x^q}.$$

Recently, Barcucci, Bernini, Bilotta and Pinzani \cite{Barc} extended this bijection to $q$-decreasing words for any positive rational number $q$, showing that $\mathcal{W}^q$ is in one-to-one correspondence with binary words avoiding some patterns.

These words have also been studied from a generative prospective. Baril et al. \cite{bkv gray code} provide efficient algorithms for the generation of all $q$-decreasing words whenever $q$ is a positive integer. In particular, they construct a $3$‑Gray code for general $q$, and notably a $1$‑Gray code for the case $q=1$, thus resolving a conjecture posed in the context of interconnection networks by \cite{egecioglu irsic}. More recently, Wong et al. \cite{Wong} present a two-stage algorithm for generating cyclic $2$-Gray codes for $q$-decreasing words.

More generally, for any $q>0$, the generating function $W_q(x)$ for the number of $q$-decreasing words with respect to the length $n$ is given by
\begin{equation}\label{eq:w_q general}
W_q(x)=\frac{1}{(1-x)\left(1-\sum_{i=0}^{+\infty}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}\right)},
\end{equation}
see \cite{sergey2,Serg}.
This expression can be simplified as follows when $q$ is rational, i.e. $q=c/d$ where $c$ and $d$ are positive integers:
\begin{equation}\label{eq:w_q rational}
W_q(x)=\frac{1-x^{c+d}}{(1-x)\left(1-x^{c+d}-\sum_{i=0}^{c-1}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}\right)}.
\end{equation}
Note that when $q$ is an integer, i.e. when we fix $d=1$ and $c=q$ in the previous formula,  we obtain $W_q(x)=\frac{F_{q+1}(x)-1}{x}$.  Using (\ref{eq:w_q general}) and (\ref{eq:w_q rational}), Dovgal and Kirgizov \cite{sergey2} proved that for all real $q>0$, $[x^n]W_q(x)\underset{n\to\infty}{\sim}C_q\cdot\Phi(q)^n$, for a positive constant $C_q$, and a function $\Phi(q)$ that interpolates the $q$-bonacci numbers. In particular, when $q=c/d$ is a rational number, $\Phi(q)^{-1}$ is the smallest root in modulus of the polynomial $x^{c+d}+\sum_{i=0}^{c-1}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}-1$. See \cite{sergey2} for additional properties of $\Phi(q)$.

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$,  then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
  meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
  An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
  $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An  element $w$ is said to {\it cover} an element $v$  if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write  $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

In this paper, we equip $\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\ldots v_n$ and  $w=w_1w_2\ldots w_n$ are two $q$-decreasing words in $\mathcal{W}_n^q$ then
$$v\leq w\Longleftrightarrow v_i\leq w_i \mbox{ for all } 1\leq i\leq n.$$

Let  $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation.  See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
 $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
 with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\sum_{n\geq 0}F_n^qx^n=\frac{1}{1-x-x^2-\cdots -x^q}.$$

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$, then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
 meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
 An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
 $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An element $w$ is said to {\it cover} an element $v$ if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

In this paper, we equip $\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\ldots v_n$ and $w=w_1w_2\ldots w_n$ are two $q$-decreasing words in $\mathcal{W}_n^q$ then
$$v\leq w\Longleftrightarrow v_i\leq w_i \mbox{ for all } 1\leq i\leq n.$$

Let $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation. See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
 $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
 with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$, then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
 meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
 An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
 $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An element $w$ is said to {\it cover} an element $v$ if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

Let $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation. See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\begin{figure}[h]
 \centering
\begin{tikzpicture}[scale=1.3]
 \node (11111) at (0,7.5) {$11111$};
 \node (11110) at (-1.5,6) {$11110$};
 \node (11001) at (0,4.5) {$11001$};
 \node (11100) at (-3,4.5) {$11100$};
 \node (00011) at (3,3) {$00011$};
 \node (10001) at (1,3) {$10001$};
 \node (10010) at (-1,3) {$10010$};
 \node (11000) at (-3,3) {$11000$};
 \node (00001) at (3,1.5) {$00001$};
 \node (00010) at (1,1.5) {$00010$};
 \node (10000) at (-3,1.5) {$10000$};
 \node (00100) at (-1,1.5) {$00100$};
 \node (00000) at (0,0) {$00000$};
 \draw (00000) -- (00100);\draw (00000) -- (10000);\draw (00000) -- (00001);\draw (00000) -- (00010);\draw (10000) -- (11000);\draw (10000) -- (10010);\draw (10000) -- (10001);\draw (00010) -- (10010);\draw (00010) -- (00011);\draw (00001) -- (10001);\draw (00001) -- (00011);\draw (00100) -- (11100);\draw (11000) -- (11001);\draw (11000) -- (11100);\draw (10010) -- (11110);\draw (10001) -- (11001);\draw (00011) -- (11111);\draw (11100) -- (11110);\draw (11001) -- (11111);\draw (11110) -- (11111);
\end{tikzpicture}
 \caption{The lattice $\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }
 \label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \( q > 0 \), we provide a closed-form expression for the generating function counting the number of coverings in \( \W^q_n \). When \( q \) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\Phi(q)$ defined in the introduction above. In Section~\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \( \W^q_n \) for any rational number \( q > 0 \). Finally, Section~\ref{sec:meet} presents the structure of meet-irreducible elements in $\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\W_n^q$ for any rational number $q>0$.

In this section, we provide enumerative results for classical parameters of a lattice, namely the join-irreducible elements, as well as covering relations. The enumeration of meet-irreducible elements is a bit more intricate, so we treat it in Section \ref{sec:meet}.
We first give the enumeration of join-irreducible elements for any $q>0$, and then we conclude by giving closed form for the generating functions of the covering for positive rational numbers $q$, and a method for computing arbitrarily many terms of the generating functions for positive irrational numbers $q$.
\subsection{Join-irreducible elements}
\begin{thm}\label{join irreducible}
 For $q>0$ and $n\geq1$, there are exactly $n$ join-irreducible elements in $\W_n^q$.
\end{thm}
\begin{proof}
 In a finite lattice, an element is join-irreducible if and only if it covers exactly one element. We therefore count the words in $\W_n^q$ that cover exactly one element. Note that a factor $0^a1^b$, with $qa>b\geq 1$ covers only one element if and only if $b=1$ (otherwise it covers $0^a1^{b-1}0$ and $0^{a+1}1^{b-1}$). Now we investigate the elements covered by $1^m$. Given $i\geq0$, there is at most one word covered by $1^m$ with suffix $01^i$. So the words covered by $1^m$ are exactly

\begin{proof} For any words $v$ and $w$ in $\mathcal{W}_n^q$, we consider the binary word $a=a_1\ldots a_n$  where $a_i=1$ if and only if $v_i=w_i=1$. It is straightforward to see that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$,  which implies that $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.
\end{proof}

\begin{figure}[h]
    \centering
\begin{tikzpicture}[scale=1.3]
    \node (11111) at (0,7.5) {$11111$};
    \node (11110) at (-1.5,6) {$11110$};
    \node (11001) at (0,4.5) {$11001$};
    \node (11100) at (-3,4.5) {$11100$};
    \node (00011) at (3,3) {$00011$};
    \node (10001) at (1,3) {$10001$};
    \node (10010) at (-1,3) {$10010$};
    \node (11000) at (-3,3) {$11000$};
    \node (00001) at (3,1.5) {$00001$};
    \node (00010) at (1,1.5) {$00010$};
    \node (10000) at (-3,1.5) {$10000$};
    \node (00100) at (-1,1.5) {$00100$};
    \node (00000) at (0,0) {$00000$};
    \draw (00000) -- (00100);\draw (00000) -- (10000);\draw (00000) -- (00001);\draw (00000) -- (00010);\draw (10000) -- (11000);\draw (10000) -- (10010);\draw (10000) -- (10001);\draw (00010) -- (10010);\draw (00010) -- (00011);\draw (00001) -- (10001);\draw (00001) -- (00011);\draw (00100) -- (11100);\draw (11000) -- (11001);\draw (11000) -- (11100);\draw (10010) -- (11110);\draw (10001) -- (11001);\draw (00011) -- (11111);\draw (11100) -- (11110);\draw (11001) -- (11111);\draw (11110) -- (11111);
\end{tikzpicture}
    \caption{The lattice $\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }
    \label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \( q > 0 \), we provide a closed-form expression for the generating function counting the number of coverings in \( \mathbb{W}^q_n \). When \( q \) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\Phi(q)$ defined in the introduction above. In Section~\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \( \mathbb{W}^q_n \) for any rational number \( q > 0 \). Finally, Section~\ref{sec:meet} presents the structure of meet-irreducible elements in $\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet  of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\W_n^q$ for any rational number $q>0$.

For any words $v$ and $w$ in $\mathcal{W}_n^q$, consider the binary word $a=a_1\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is \"straightforward to see\" that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.

For any words $v$ and $w$ in $\mathcal{W}_n^q$, consider the binary word $a=a_1\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is "straightforward to see" that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.,

Question: Let \(q\ge 0\) and \(n\ge 1\). A binary word of length \(n\) is called \(q\)-decreasing if every maximal factor of the form \(0^a1^b\) satisfies either \(a=0\) or \(q\,a>b\). Let \(\mathcal{W}_n^q\) be the set of all such words, and equip it with the componentwise order: for \(v=v_1\cdots v_n\) and \(w=w_1\cdots w_n\), define \(v\le w\) iff \(v_i\le w_i\) for all \(1\le i\le n\). Write \(\W_n^q=(\mathcal{W}_n^q,\le)\). Which statement holds for every choice of \(q\ge 0\) and \(n\ge 1\)?

The poset \(\W_n^q\) is a lattice; equivalently, every pair of elements of \(\mathcal{W}_n^q\) has both a meet (greatest lower bound) and a join (least upper bound) in the componentwise order.

Choices:
B. The poset \(\W_n^q\) is a meet-semilattice but, for some choices of \(q\ge 0\) and \(n\ge 1\), it is not a lattice; in general, componentwise meets of elements of \(\mathcal{W}_n^q\) always exist, whereas componentwise joins need not belong to \(\mathcal{W}_n^q\).
C. For every \(q\ge 0\) and \(n\ge 1\), every pair of elements of \(\mathcal{W}_n^q\) has a meet (greatest lower bound) in the componentwise order; equivalently, \(\W_n^q\) is a meet-semilattice.
D. The poset \(\W_n^q\) is a lattice for every irrational \(q>0\), but this can fail for rational values of \(q\) because the distinction between the strict condition \(q\,a>b\) and the weak condition \(q\,a\ge b\) prevents componentwise joins from being well defined in general.
E. The poset \(\W_n^q\) is a lattice precisely for \(q>0\); when \(q=0\), the condition on maximal factors forces a boundary case in which \(\mathcal{W}_n^q\) is no longer closed under the meet and join operations in the componentwise order.

The stem defines the objects and order relation but does not explicitly or implicitly reveal that the poset is a lattice. No wording strongly privileges choice A over the others.

The item is close to a theorem-identification question: the correct option is essentially a clean theorem statement about \(\W_n^q\). It is not a pure tautology because the alternatives offer nearby competing claims, but it still largely tests recognition of the main result.

There is some reasoning pressure because the student must distinguish lattice vs. meet-semilattice, universal vs. exceptional-in-\(q\) behavior, and boundary cases such as \(q=0\). However, the item mainly rewards theorem recall or result recognition rather than substantial derivation from first principles.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, others exploit common failure modes involving joins, rational/irrational distinctions, and the boundary case \(q=0\). They are distinct and well targeted.

A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward recall of the exact result than toward genuinely generative mathematical reasoning.


Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\mathbb R[x_1,\dots,x_n],$ contains more simple modules than $\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\mathbb C$-algebra $A_{\mathbb R}$ such that $\simp(A_{\mathbb R})\cong\mathbb R^n.$ Thus the points in $\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\mathbb R}$-modules for which $\aspec(A_{\mathbb R})$ is a fine moduli, see the book \cite{S23} or the preprint \cite{S241}. For any real manifold $M$ we define an associative variety $(\mathcal M,\mathcal O^A_L)$ over $\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\mathcal M,$ and such that the charts $U$ in $\mathcal M$ corresponds to $\mathcal O^A_L(U)=A_{\mathbb R}.$

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

Because of the discovery of localization in associative rings, \cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.
Let $k$ be a field and consider the polynomial algebra in $n\geq 1$ variables $$A=k[x_1,\dots,x_n]=k[\underline x].$$ For a point $P=(p_1,\dots,p_n)\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\dots,x_n-p_n).$$

\begin{abstract}
In the preprint \cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\rightarrow B$ we define the contraction of a simple $B$-module to $A.$
Then we define the set of aprime right $A$-modules $\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\aspec A=\spec A,$ and we define a topology on $\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\mathcal O_X$ on $\aspec A,$ agreeing with the usual sheaf of rings on $\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\subseteq\aspec A.$ In particular, this
proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\mathbb R\subseteq\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.
\end{abstract}

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

\begin{proof} We have a homomorphism $f:A\rightarrow A/\mathfrak m$ such for every $s\in A\setminus\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\phi:A_\mathfrak m\rightarrow A/\mathfrak m$ such that $\phi(\frac{a}{s})=\iota(a)\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\mathfrak m A_\mathfrak m$ giving the wanted isomorphism.
\end{proof}

\begin{proof} We send the prime ideal $\mathfrak p\subset A$ to $A_\mathfrak p/\mathfrak p A_\mathfrak p$ which is $A$-prime by definition. On the other hand, let
$M$ be an aprime $A$-module, defined by $\iota_M: A\rightarrow B$ such that $M$ is a simple $B$-module. Let $\mathfrak m\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\iota_M^{-1}(\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\mathfrak m\simeq B_{\mathfrak m}/\mathfrak m B_\mathfrak m$ as proven i Lemma \ref{anotherloclemma}.
\end{proof}

\begin{definition} Define the subset of $k$-points in $\mathbb X$ by $$\tilde{\mathbb X}(k)=\{x\in X\subseteq\mathbb X|x\text{ is simple}\}\subseteq\mathbb X.$$ Then the induced associative subscheme $\mathbb X(k)$ is called the associative subscheme of $k$-points.
\end{definition}

\begin{proof} Because $k\subseteq\Bbbk$ is a sub-algebra, it follows that if $\phi\otimes\id$ is an isomorphism, then $\dim_k V_1=\dim_k V_2,$ and that choosing corresponding bases, $$0\neq\det(\phi\otimes\id)=\det\phi.$$
\end{proof}

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P=Q$ we can assume $M_1=M_2=k[\underline x]/(\underline x).$ Then any inner derivation is on the form $\ad_\phi$ for which $\ad_\phi(x_i)=[\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\ext^1_A(M_1,M_2)\simeq\der_k(A,k)$ where
$\operatorname d_i(x_i)=1,\ i=1,\dots,n$ gives a basis for the derivations.

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

It follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \cite{S23} for a lot of examples.

From \cite{S23}: \(\ext^1_A(M_1,M_2)\simeq \der_k(A,\hmm_k(M_1,M_2))/\inner\), with \(\hmm_k(M_1,M_2)\) an \(A\)-\(A\) bimodule via \(a\phi(m)=\phi(am)\), \(\phi a(m)=\phi(m)a\).

- Case \(P=Q\): assume \(M_1=M_2=k[\underline x]/(\underline x)\). Any inner derivation is \(\ad_\phi\) and \(\ad_\phi(x_i)=[\phi,x_i]=0\), so \(\dim_k\inner=0\). Hence \(\ext^1_A(M_1,M_2)\simeq \der_k(A,k)\), and the derivations \(\operatorname d_i\) with \(\operatorname d_i(x_i)=1\) for \(i=1,\dots,n\) give a basis, so the dimension is \(n\).

- Case \(P\neq Q\): consider \(M_{\underline 0}, M_P\) with \(p_1\neq 0\). For \(\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P))\), comparing \(\delta(x_ix_1)=\delta(x_1x_i)\) yields
\[\delta(x_i)x_1+x_i\delta(x_1)=\delta(x_1)x_i+x_1\delta(x_i)\Rightarrow \delta(x_i)p_1=\delta(x_1)p_i\Rightarrow \delta(x_i)=\delta(x_1)\frac{p_i}{p_1}.\]
This relation also holds for inner derivations, and it is concluded that \(\dim_k\inner=1\), so \(\ext^1_A(M_P,M_Q)=0\) when \(P\neq Q\).

For two points $P,Q\in k^n$ we have that $$\dim_k\ext^1_A(M_P,M_Q)=\begin{cases}n,\ P=Q\\0,\ P\neq Q.\end{cases}$$,

Question:
Let $k$ be a field, let $A=k[x_1,\dots,x_n]$ be the polynomial algebra in $n\ge 1$ variables over $k$, and for each point $P=(p_1,\dots,p_n)\in k^n$ define the simple $A$-module
$$M_P:=A/(x_1-p_1,\dots,x_n-p_n).$$
Which statement holds for every pair of points $P,Q\in k^n$ about the $k$-dimension of $\operatorname{Ext}^1_A(M_P,M_Q)$?

$$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\ne Q.\end{cases}$$

Choices:
B. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\1,& P\ne Q.\end{cases}$$
C. $$\dim_k\operatorname{Ext}^1_A(M_P,M_P)=n\qquad\text{for every }P\in k^n.$$
D. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\text{ and }Q\text{ differ in every coordinate}.\end{cases}$$
E. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n-1,& P=Q,\\0,& P\ne Q.\end{cases}$$

quotient by inner derivations in the off-diagonal case

use of a coordinate with nonzero difference after reducing to one point at the origin

dimension of derivations in the diagonal case

The stem does not state or strongly hint at the diagonal/off-diagonal Ext^1 dimensions; it only defines the modules and asks for the correct universal statement.

This is not a mere restatement of a theorem from the stem. The respondent must choose among several competing global descriptions, including a weaker true statement and subtly false variants.

Answering correctly requires genuine reasoning or recall of the Ext computation: one must identify both the diagonal value n and the off-diagonal vanishing, and also recognize that choice C is true but not the strongest statement asked for.

The distractors are plausible and mathematically targeted: B tests mistaken off-diagonal nonvanishing, C is a tempting weaker true statement, D reflects an over-specific coordinate-based vanishing claim, and E probes a dimension-count error.

A strong MCQ: no answer leakage, non-tautological framing, real reasoning pressure, and high-quality distractors that reflect common mathematical missteps.