[Hirose--Sato~\cite{HiroseSato2019A}]\label{thm:HiroseSato1}
For positive integers $a, b$, and $c$, we have
\begin{align*}
&\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)\\
&=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\end{align*}

\begin{theorem}[Hirose--Maesaka--Seki--Watanabe~\cite{HiroseMaesakaSekiWatanabe2025+}]
The drop 1 operator $\cD$ satisfies the following properties$:$
\label{thm:drop1}
\begin{enumerate}[label=\textnormal{(\roman*)}]
\item\label{it:HMSW1} $\{w-\cD(w) \mid w\in \cH^0\}=\Ker(Z^{\diamondsuit})\subset\Ker(Z)$,
\item\label{it:HMSW2} $\cD(w)=w$ for $w\in\cH^{\geq 2}$,
\item\label{it:HMSW3} $\cD(w)=\cD(\tau(w))$ for $w\in \cH^0$.
\end{enumerate}
\end{theorem}

\begin{definition}[Hirose--Maesaka--Seki--Watanabe~\cite{HiroseMaesakaSekiWatanabe2025+}]\label{def:drop1operator}
We define the \emph{drop 1 operator} $\cD\colon\cH^0\to\cH^{\geq 2}$ to be a $\ZZ$-linear map given as follows.
It suffices to define $\cD(w)$ for each word $w\in\cH^0$.
For $\bc=(c_1,\dots, c_{2t})$, we write $\fD(\bc)=\cD(y^{c_1}x^{c_2}\cdots y^{c_{2t-1}}x^{c_{2t}})$.
The image $\fD(\bc)$ is defined inductively as follows:
\begin{align*}
\fD(\bc)&=\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 1}}(-1)^{\#B-1}\fD(\bc_{(-A)}-\bdelta_B)x^{\#A+\#B}\\
&\quad+\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 2}}(-1)^{\#B-1}\fD(\bc_{(-A)}-\bdelta_B)z_{\#A+\#B}\\
&\quad+\sum_{\substack{A\in\cP^{\text{oe}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{oe}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 2}}(-1)^{\#B}\fD(\bc_{(-A)}-\bdelta_B)z_{\#A+\#B}.
\end{align*}
Here the initial condition is $\fD(\varnothing)=\cD(1)=1$.
\end{definition}

\begin{corollary}\label{cor:diamond-lift}
For $w_1, w_2\in\cH^{2,3}$, we have
\[
Z^{\diamondsuit}(w_1\tau(w_2))=Z^{\diamondsuit}(w_1\star w_2).
\]
\end{corollary}

For positive integers $k_1, \dots, k_r$ with $k_r\geq 2$, the \emph{multiple zeta value} (MZV) $\zeta(k_1,\dots, k_r)$ is defined by
\[
\zeta(k_1,\dots, k_r)\coloneqq\sum_{0<n_1<\cdots <n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
Linear relations among MZVs have been studied for many years.
For each positive integer $c$, the relation
\begin{equation}\label{eq:Hoffman-conj}
\zeta(3,\{2\}^{c-1},1,2)=\zeta(\{2\}^{c+2})+2\zeta(3,3,\{2\}^{c-1})
\end{equation}
is known as \emph{Hoffman's conjectural identity} and had remained open for about seventeen years.
Here the notation $\{2\}^n$ denotes the string $2,\dots, 2$ in which the entry 2 is repeated $n$ times.
Charlton proved in \cite{Charlton2016} that there exists a rational number $q_c$ such that $\zeta(3,\{2\}^{c-1},1,2)=q_c\zeta(\{2\}^{c+2})+2\zeta(3,3,\{2\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.

More generally, applying the same procedure to Corollary~\ref{cor:diamond-lift} (see Section~\ref{sec:finite} for more details), we have the following.
For a word $w=z_{k_1}\cdots z_{k_r}$, we set $\overleftarrow{w}\coloneqq z_{k_r}\cdots z_{k_1}$, and $\wt(w)$ denotes the total degree of $w$, that is, $\wt(w)=k_1+\cdots + k_r$.
A $\ZZ$-linear map $Z_{\cA}\colon \ZZ+y\cH=\ZZ\langle z_1,z_2,\dots\rangle\to\cA$ is defined by $Z_{\cA}(z_{k_1}\cdots z_{k_r})=\zeta^{}_{\cA}(k_1,\dots, k_r)$ and $Z_{\cA}(1)=1$.
\begin{corollary}\label{cor:FMZV}
For words $w_1, w_2\in\cH^{2,3}$, we have
\[
(-1)^{\wt(w_2)}Z_{\cA}(w_1\overleftarrow{w_2})=Z_{\cA}(w_1\star w_2).
\]
\end{corollary}
This is a family of relations among FMZVs in which only 2 and 3 occur as entries.
Since the shuffle relation for FMZVs $(-1)^{\wt(w_2)}Z_{\cA}(w_1\overleftarrow{w_2})=Z_{\cA}(w_1\sh w_2)$ holds (see \cite{KanekoZagier2025+,Ono2017}), Corollary~\ref{cor:FMZV} can also be rewritten as
\[
Z_{\cA}(w_1\sh w_2-w_1\star w_2)=0
\]
which is of the form of a certain double shuffle relation.
\section*{Acknowledgements}
The author would like to thank Professor Minoru Hirose for kindly answering his questions about the previous works by Hirose and Sato.
This research was inspired by a question that came to mind when the author got married, and the author would like to thank his wife, Yui, for marrying him.
\section{The drop 1 operator}
We introduce the following notation to define the main operator.
Let $t$ be a positive integer, let $\bc=(c_1,\dots, c_{2t})$ be a tuple of positive integers, and let $S\subset\ZZ$.
\begin{itemize}[leftmargin=2.5em]
\item $\#A$ denotes the cardinality of a finite set $A$.
\item $[n]\coloneqq\{1,2,\dots,n\}$ for $n\in\ZZ_{\geq 0}$. $[0]=\varnothing$.
\item $[2t]_{\bc}^1\coloneqq\{i\in[2t] \mid c_i=1\}.$
\item $[2t]_{\bc}^{>1}\coloneqq\{i\in[2t] \mid c_i>1\}.$
\item $\cP^{\text{eo}}(S)\coloneqq\{A\subset S \mid i\in A\cap 2\ZZ \Longleftrightarrow i+1\in A\setminus 2\ZZ\}.$
\item $\cP^{\text{oe}}(S)\coloneqq\{A\subset S \mid i\in A\setminus 2\ZZ \Longleftrightarrow i+1\in A\cap 2\ZZ\}.$
\item $\cP^{\smallxcancel{\text{eo}}}(S)\coloneqq\{A\subset S \mid i\in A\cap 2\ZZ \Longrightarrow i+1\not\in A\setminus 2\ZZ\}.$
\item $\cP^{\smallxcancel{\text{oe}}}(S)\coloneqq\{A\subset S \mid i\in A\setminus 2\ZZ \Longrightarrow i+1\not\in A\cap 2\ZZ\}.$
\item $\bc_{(-A)}\coloneqq (c_{i_1},\dots, c_{i_r})$, where $[2t]\setminus A=\{i_1,\dots, i_r\}$ with $i_1<\cdots<i_r$, for $A\subset[2t]$.
\item $\bc_{(-\varnothing)}=\bc$, $\bc_{(-[2t])}=\varnothing$.
\item $\delta_{i\in B}\coloneqq\begin{cases}1 & \text{if } i\in B, \\ 0 & \text{otherwise}\end{cases}$ for a set $B$.
\item $\bc_{(-A)}-\bdelta_B\coloneqq(c_{i_1}-\delta_{i_1\in B},\dots, c_{i_r}-\delta_{i_r\in B})$ for the above $\bc_{(-A)}$ and $B\subset [2t]$.
\end{itemize}

We consider the case $c>1$.
Then $\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc})=\cP([2t]^{>1}_{\bc})$ holds.
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$, decompose it as $A=A_1\cup A_2 \cup A_3$, where
\[
A_1\coloneqq A \cap [2l_r-1],\quad A_2\coloneqq A\cap ([2l_r+2c-2]\setminus [2l_r]),\quad A_3\coloneqq A\setminus [2l_r+2c-1].
\]
We define a map $\iota_1\colon\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ by
\[
\iota_1(A)\coloneqq (A_1-1)\cup (A_2-1)\cup\{2l_r+2c-3,2l_r+2c-2\}\cup(A_3+1)
\]
and a map $\iota_2\colon\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ by
\[
\iota_2(A)\coloneqq (A_1-1)\cup (A_2-1)\cup(A_3+1).
\]
A bijection $\cP^{\text{eo}}([2t]^1_{\bc})\sqcup\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ is obtained by sending an element $A$ in the first copy of $\cP^{\text{eo}}([2t]^1_{\bc})$ to $\iota_1(A)$, and an element $A$ in the second copy of $\cP^{\text{eo}}([2t]^1_{\bc})$ to $\iota_2(A)$.
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$, we have $\bc_{(-A)}=\bc_{(-\iota_2(A))}$.
Therefore, if we compute the defining recurrence for the drop 1 operator using this bijection, every term in the second sum is canceled by a term in the third sum, and we are left with
\begin{align*}
\fD(\bc)&=\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 1}}(-1)^{\#B-1}\biggl(\fD(\bc_{(-A)}-\bdelta_B)-\fD(\bc_{(-\iota_1(A))}-\bdelta_B)z_2\biggr)x^{\#A+\#B}\\
&\quad+\fD(\bc_{(-\{2l_r+2c-3,2l_r+2c-2\})})z_2.
\end{align*}
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$ or $\cP^{\text{oe}}([2t]^1_{\bc})$, there exist $a'_1$, $\dots$, $a'_r$, $b'_1$, $\dots$, $b'_s$, and $c'$ such that
\[
\bc_{(-A)}=[a'_1,\dots, a'_r;b'_1,\dots, b'_s;c'].
\]
Here $a'_1\in[a_1]$, $\dots$, $a'_r\in[a_r]$, $b'_1\in[b_1]$, $\dots$, $b'_s\in[b_s]$, $c'\in[c]$, and $a'_1+\cdots+a'_r+b'_1+\cdots+b'_s+c'=t+1-\#A/2$.
Moreover, upon adding $-\bdelta_B$, addition occurs between the corresponding components; for example, when
\begin{align*}
\bc&=[2,1,3;4,1,1,2;3]\\
&=(\stackrel{1}{1},\stackrel{2}{1},\stackrel{3}{1},\stackrel{\color{red}{4}}{2},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{8}{1},\stackrel{9}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{14}{1},\stackrel{15}{1},\stackrel{16}{1},\stackrel{\color{red}{17}}{2},\stackrel{18}{1},\stackrel{19}{1},\stackrel{20}{1},\stackrel{21}{1},\stackrel{22}{1},\stackrel{23}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{red}{27}}{2},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{30}{1},\stackrel{31}{1},\stackrel{32}{1}),
\end{align*}
\[
A=\{2,3,8,9,14,15,18,19,22,23,30,31\}\in\cP^{\text{eo}}([32]^1_{[2,1,3;4,1,1,2;3]}),
\]
and
\[
B=\{4,17,27\}\subset[32]^{>1}_{[2,1,3;4,1,1,2;3]}=\{4,6,12,17,25,27,29\},
\]
we have
\begin{align*}
&[2,1,3;4,1,1,2;3]_{(-\{2,3,8,9,14,15,18,19,22,23,30,31\})}=[1,1,2;2,1,1,1;2]\\
&=(\stackrel{1}{1},\stackrel{\color{red}{4}}{2},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{16}{1},\stackrel{\color{red}{17}}{2},\stackrel{20}{1},\stackrel{21}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{red}{27}}{2},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{32}{1})
\end{align*}
and
\begin{align*}
&[2,1,3;4,1,1,2;3]_{(-\{2,3,8,9,14,15,18,19,22,23,30,31\})}-\bdelta_{\{4,17,27\}}\\
&=[1+1,2;1+1,1;2+2]=[2,2;2,1;4]\\
&=(\stackrel{1}{1},\stackrel{\color{blue}{4}}{1},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{16}{1},\stackrel{\color{blue}{17}}{1},\stackrel{20}{1},\stackrel{21}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{blue}{27}}{1},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{32}{1}).
\end{align*}
Therefore, for any $A, B$ with $\#A+\#B\geq 1$, the induction hypothesis implies that there exist words $v_1, v_2$ in $\cH^{2,3}$ such that $\bc_{(\iota_1(-A))}-\bdelta_B$ corresponds to $v_1\tau(v_2)$, and we have
\[
\fD(\bc_{(-A)}-\bdelta_B)-\fD(\bc_{(-\iota_1(A))}-\bdelta_B)z_2=\cD(v_1z_2\tau(v_2))-\cD(v_1\tau(v_2))z_2=v_1z_2\star v_2-(v_1\star v_2)z_2=0.
\]
Since we are in the case $c>1$, we can write $w_1=wz_2$ with a certain word $w$ in $\cH^{2,3}$.
Then we obtain
\begin{align*}
\fD(\bc)&=\fD(\bc_{(-\{2l_r+2c-3,2l_r+2c-2\})})z_2=\fD([a_1,\dots, a_r;b_1,\dots, b_s;c-1])z_2=\cD(w\tau(w_2))z_2\\
&=(w\star w_2)z_2=wz_2\star w_2=w_1\star w_2.
\end{align*}

Their proof can be viewed as showing that the above identity is a consequence of the \emph{confluence relation} later formulated in \cite{HiroseSato2019B}.

Hirose and Sato gave a further generalization in \cite{HiroseSato2022+}, and we set up notation in order to state their result.
Let $\mathcal{H}\coloneqq\mathbb{Z}\langle x,y\rangle$ be the non-commutative polynomial ring over $\mathbb{Z}$ in two variables $x$ and $y$.
Let $z_k\coloneqq yx^{k-1}$ for each positive integer $k$.
Subrings $\mathcal{H}^{2,3}\subset\mathcal{H}^0\subset\mathcal{H}$ are defined by $\mathcal{H}^{2,3}\coloneqq\mathbb{Z}\langle z_2,z_3\rangle$ and $\mathcal{H}^0\coloneqq\mathbb{Z}+y\mathcal{H} x$, respectively.
A $\mathbb{Z}$-linear map $Z\colon\mathcal{H}^0\to\mathbb{R}$ is defined by $Z(z_{k_1}\cdots z_{k_r})=\zeta(k_1,\dots, k_r)$ and $Z(1)=1$.
A product $\star$ on $\mathcal{H}^{2,3}$ is defined inductively by
\begin{itemize}[leftmargin=2.5em]
\item $w\star 1=1\star w=w$,
\item $wz_2\star w'=w\star w'z_2=(w\star w')z_2$,
\item $wz_3\star w'z_3=(w\star w'z_3)z_3+(wz_3\star w')z_3+(w\star w')z_2^3$
\end{itemize}
for any words $w$ and $w'$ in $\mathcal{H}^{2,3}$, and then extended by $\mathbb{Z}$-bilinearity.
Let $\tau$ be an anti-automorphism on $\mathcal{H}$ defined by $\tau(x)=y$ and $\tau(y)=x$.
Using this notation, we can now state the following theorem.
\begin{theorem}[{Hirose--Sato~\cite[Proposition~14]{HiroseSato2022+}}]\label{thm:HiroseSato2}
For $w, w'\in\mathcal{H}^{2,3}$, we have
\[
Z(w_1\tau(w_2))=Z(w_1\star w_2).
\]
\end{theorem}
By specializing Theorem~\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$, we recover Theorem~\ref{thm:HiroseSato1}.
Hirose and Sato derived Theorem~\ref{thm:HiroseSato2} from their \emph{block shuffle identity} (\cite[Theorem~10]{HiroseSato2022+}).

Hoffman conjectured in \cite{Hoffman1997} that any multiple zeta value can be expressed as a $\mathbb{Q}$-linear combination of the elements of
\[
\{\zeta(k_1,\dots, k_r) \mid r\in\ZZ_{>0}, k_1,\dots, k_r\in\{2,3\}\}.
\]
This conjecture was proved by Brown in \cite{Brown2012}.
Although the linear independence of these elements remains open, they are called the \emph{Hoffman basis}.
Theorem~\ref{thm:HiroseSato2} provides a family for which the expansion in the Hoffman basis can be described explicitly, and, even more remarkably, it shows that the expansion can be taken with \emph{integer coefficients}.
In general, the coefficients are not integral; for instance,
\[
\zeta(3,1,4)=-\frac{2048}{4125}\zeta(2,2,2,2)-\frac{17}{275}\zeta(2,3,3)+\frac{2}{275}\zeta(3,2,3)+\frac{113}{275}\zeta(3,3,2).
\]

In contrast, Hirose, Maesaka, Seki, and Watanabe recently proved in \cite{HiroseMaesakaSekiWatanabe2025+}, together with an explicit algorithm, that any multiple zeta value can be expressed as a $\mathbb{Z}$-linear combination of
\[
\{\zeta(k_1,\dots, k_r) \mid r\in\ZZ_{>0}, k_1,\dots, k_r\geq 2\}.
\]
Their explicit expansions give rise to a family of relations among MZVs, which we refer to as the \emph{drop 1 relation}.
Since their expansions have integer coefficients, it is natural to ask whether Theorem~\ref{thm:HiroseSato2} can be viewed as an expansion arising from the drop 1 relation.
However, in general, expansions coming from the drop 1 relation involve MZVs with entries greater than $3$, so it is not obvious whether the expansion in Theorem~\ref{thm:HiroseSato2}, in which only 2 and 3 appear, can be obtained from the drop 1 relation.
The main result of this paper is that this is indeed the case.

A subring $\mathcal{H}^{\geq 2}\subset\mathcal{H}^0$ is defined by $\mathcal{H}^{\geq 2}\coloneqq\mathbb{Z}+z_2\mathbb{Z}\langle x,z_2\rangle=\mathbb{Z}\langle z_2, z_3, z_4,\dots\rangle$.
A linear map $\mathcal{D}\colon\mathcal{H}^0\to\mathcal{H}^{\geq 2}$, called the \emph{drop 1 operator}, is defined in Definition~\ref{def:drop1operator}, and its properties are summarized in Theorem~\ref{thm:drop1}.
\begin{theorem}\label{thm:main}
For $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
\mathcal{D}(w_1\tau(w_2))=w_1\star w_2.
\]
\end{theorem}
Not only does this theorem yield a new proof of Theorem~\ref{thm:HiroseSato2}, but it also implies, by Theorem~\ref{thm:drop1}~\ref{it:HMSW1}, that the same relations hold for \emph{multiple zeta diamond values}.
We omit the definition of $\zeta^{\diamondsuit}(k_1,\dots, k_r)$ here; see \cite[Definition~2.3]{HiroseMaesakaSekiWatanabe2025+}.
A $\mathbb{Z}$-linear map $Z^{\diamondsuit}\colon\mathcal{H}^0\to\mathbb{Q}^{\mathbb{N}}$ is defined by $Z^{\diamondsuit}(z_{k_1}\cdots z_{k_r})=\zeta^{\diamondsuit}(k_1,\dots, k_r)$ and $Z^{\diamondsuit}(1)=1$.

\begin{corollary}\label{cor:diamond-lift}
For $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
Z^{\diamondsuit}(w_1\tau(w_2))=Z^{\diamondsuit}(w_1\star w_2).
\]
\end{corollary}
For example, the relation corresponding to \eqref{eq:Hoffman-conj},
\[
\zeta^{\diamondsuit}(3,\{2\}^{c-1},1,2)=\zeta^{\diamondsuit}(\{2\}^{c+2})+2\zeta^{\diamondsuit}(3,3,\{2\}^{c-1})
\]
means that, for any positive integer $N$,
\begin{align*}
&\sum_{0<n_1<n_2<\cdots<n_{c}<n_{c+1}<n_{c+2}<N}\frac{1}{n_1^3n_2^2\cdots n_{c}^2n_{c+1}^{}n_{c+2}^2}\\
&+\sum_{0<n_1<n_2<\cdots<n_{c}<n_{c+1}\leq n_{c+2}<N}\frac{1}{n_1^3n_2^2\cdots n_{c}^2(N-n_{c+1}^{})n_{c+2}^2}\\
&=\sum_{0<n_1<\cdots < n_{c+2}<N}\frac{1}{n_1^2\cdots n_{c+2}^2}+2\sum_{0<n_1<n_2<n_3<\cdots<n_{c+1}<N}\frac{1}{n_1^3n_2^3n_3^2\cdots n_{c+1}^2}
\end{align*}
holds.
Letting $N\to\infty$ yields \eqref{eq:Hoffman-conj}.
On the other hand, if we take $N=p$ for a prime $p$ and take modulo $p$, then we have the following relation among \emph{finite multiple zeta values} (FMZVs):
\[
\zeta^{}_{\mathcal{A}}(3,\{2\}^{c-1},3)+2\zeta^{}_{\mathcal{A}}(3,3,\{2\}^{c-1})=0.
\]
Here, for positive integers $k_1,\dots, k_r$, the FMZV $\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)$ is defined as an element of $\mathcal{A}\coloneqq\left.\prod_p\mathbb{Z}/p\mathbb{Z}\right/\bigoplus_p\mathbb{Z}/p\mathbb{Z}$ by
\[
\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)\coloneqq (\zeta^{}_p(k_1,\dots,k_r)\bmod{p})_p,
\]
where $p$ runs over all prime numbers, and
\[
\zeta^{}_p(k_1,\dots,k_r)\coloneqq\sum_{0<n_1<\cdots<n_r<p}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
This definition is due to Kaneko and Zagier (\cite{KanekoZagier2025+}).
We have used a well-known formula: $\zeta^{}_{\mathcal{A}}(\{2\}^{c+2})=0$.

More generally, applying the same procedure to Corollary~\ref{cor:diamond-lift} (see Section~\ref{sec:finite} for more details), we have the following.
For a word $w=z_{k_1}\cdots z_{k_r}$, we set $\overleftarrow{w}\coloneqq z_{k_r}\cdots z_{k_1}$, and $\mathrm{wt}(w)$ denotes the total degree of $w$, that is, $\mathrm{wt}(w)=k_1+\cdots + k_r$.
A $\mathbb{Z}$-linear map $Z_{\mathcal{A}}\colon \mathbb{Z}+y\mathcal{H}=\mathbb{Z}\langle z_1,z_2,\dots\rangle\to\mathcal{A}$ is defined by $Z_{\mathcal{A}}(z_{k_1}\cdots z_{k_r})=\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)$ and $Z_{\mathcal{A}}(1)=1$.
\begin{corollary}\label{cor:FMZV}
For words $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
(-1)^{\mathrm{wt}(w_2)}Z_{\mathcal{A}}(w_1\overleftarrow{w_2})=Z_{\mathcal{A}}(w_1\star w_2).
\]
\end{corollary}
This is a family of relations among FMZVs in which only 2 and 3 occur as entries.
Since the shuffle relation for FMZVs $(-1)^{\mathrm{wt}(w_2)}Z_{\mathcal{A}}(w_1\overleftarrow{w_2})=Z_{\mathcal{A}}(w_1\sh w_2)$ holds (see \cite{KanekoZagier2025+,Ono2017}), Corollary~\ref{cor:FMZV} can also be rewritten as
\[
Z_{\mathcal{A}}(w_1\sh w_2-w_1\star w_2)=0
\]
which is of the form of a certain double shuffle relation.

Their proof (of Theorem~\ref{thm:HiroseSato1}) “can be viewed as showing that the above identity is a consequence of the \emph{confluence relation}.”

The text then indicates that Theorem~\ref{thm:HiroseSato1} is recovered by “specializing Theorem~\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$,” and that Hirose--Sato “derived Theorem~\ref{thm:HiroseSato2} from their \emph{block shuffle identity}.”

Their proof (of the main theorem) “can be viewed as showing that the above identity is a consequence of the \emph{confluence relation}.”

The text then indicates that the main theorem is recovered by specializing the following theorem to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$.

\begin{theorem}[{Hirose--Sato~\cite[Proposition~14]{HiroseSato2022+}}]\label{thm:HiroseSato2}
For $w, w'\in\cH^{2,3}$, we have
\[
Z(w_1\tau(w_2))=Z(w_1\star w_2).
\]
\end{theorem}

It is also stated that Hirose--Sato “derived the theorem above from their \emph{block shuffle identity}.”

Question:
For positive integers $k_1,\dots,k_r$ with $k_r\ge 2$, the multiple zeta value is defined by
\[
\zeta(k_1,\dots,k_r)=\sum_{0<n_1<\cdots<n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
For an integer $n\ge 0$, let $\{2\}^n$ denote the string consisting of $n$ copies of $2$ (so $\{2\}^0$ is the empty string). Which identity holds for every choice of positive integers $a,b,c$?

\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]

Choices:
B.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+2\,\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1}).
\]
C.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+R_{a,b,c},
\]
where
\[
R_{a,b,c}=\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]
D.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{c-1},3,\{2\}^{a-1}).
\]
E.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c+1})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]

symmetrized two-term contribution collapsed to a doubled single term

explicit decomposition of the remainder into the two displayed zeta terms

precise placement of the blocks in the second 3-term on the right-hand side

total weight balance between the left-hand side and the pure even zeta term

The stem only gives the definition of multiple zeta values and notation; it does not reveal or strongly hint at the specific identity in choice A.

The item is not a direct restatement of a theorem from the stem. The student must choose among several structurally similar candidate identities.

There is some reasoning/knowledge required to distinguish the precise placement, symmetry, and signs in A/B/D/E. However, choice C is a weak existential statement that is effectively easier to endorse without knowing the exact identity, which reduces genuine generative pressure.

B, D, and E are plausible theorem-variant distractors involving coefficient, sign, and block-order mistakes. But C is a poor distractor because it is a weaker true statement, making the item ambiguous rather than cleanly discriminative.

A mathematically sophisticated and non-leaky item, but weakened by an ambiguous weaker-true distractor (C), which reduces its effectiveness as a single-answer reasoning question.


A {\it dominating set} for a graph $G = (V,E)$ is a subset $S \subseteq V$ such that every vertex $v\in V$ is either in $S$ or has a neighbor $u\in S$. The vertex $u \in S$ is said to \textit{cover}  the vertex $v \in V$ if either $u=v$ or $uv\in E$. A dominating set $S$ is a \textit{minimal dominating set} if no proper subset $S'\subset S$ is a dominating set. The \textit{domination number} $\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \textbf{$\gamma$-sets}.

Dominating sets, in particular $\gamma$-sets, have been extensively researched. We recommend to the reader Haynes et al.'s book \cite{Hay2} and several other research works \cite{FB, BR, EC, GD, MH, FH, Hay1, ML} on variants and applications of dominating sets.
In this literature we found various versions of $\gamma$-set such as the perfect dominating set, the double dominating set, the strong dominating set, and the restrained dominating set. In general, each such type of $\gamma$-set has some specified properties. Here, we list five of the most commonly studied $\gamma$-sets along with their properties. Later in Example \ref{Ex}, we illustrate each such $\gamma$-set.

\begin{enumerate}
	\item \textit{Perfect $\gamma$-set} requires that no two vertices in a $\gamma$-set cover the same vertex (outside the $\gamma$-set). \item \textit{Connected $\gamma$-set} requires that the graph induced by the $\gamma$-set be connected.
	\item \textit{Total $\gamma$-set} forbids isolated vertices in the subgraph induced by the $\gamma$-set.
	\item \textit{Independent $\gamma$-set} requires all vertices in a $\gamma$-set to be isolated vertices in the subgraph they induce.
	\item \textit{clique $\gamma$ set} requires that the vertices in a $\gamma$-set induce a clique.
\end{enumerate}

Because a given graph may have multiple $\gamma$-sets, this article addresses a natural but fundamental question:\textit{ How many  $\gamma$-sets does a given graph have?} To answer this question, we introduce the notion of \textbf{dominion}.

The \textit{dominion (number)} of  a graph $G=(V,E)$, denoted by $\zeta(G)$, is the number of its $\gamma$-sets. In other words,

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a  complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
	\centering
	\begin{tikzpicture}[scale=1]
	\clip(0,-1) rectangle (4,4);
	\draw [line width=2pt] (2.62,1)-- (2,2);
	\draw [line width=2pt] (2,2)-- (1.98,2.84);
	\draw [line width=2pt] (2.62,1)-- (3,0.34);
	\draw [line width=2pt] (2,2)-- (1.48,1.02);
	\draw [line width=2pt] (1.48,1.02)-- (2.62,1);
	\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
	\begin{scriptsize}
	\draw [fill=black] (2,2) circle (2.5pt);
	\draw[color=black] (2.2,2.1) node {$u$};
	\draw [fill=black] (2.62,1) circle (2.5pt);
	\draw[color=black] (2.92,1) node {$v$};
	\draw [fill=black] (1.98,2.84) circle (2.5pt);
	\draw[color=black] (1.94,3.1) node {$x$};
	\draw [fill=black] (3,0.34) circle (2.5pt);
	\draw[color=black] (3.2,0.15) node {$y$};
	\draw [fill=black] (1.48,1.02) circle (2.5pt);
	\draw[color=black] (1.14,1) node {$w$};
	\draw [fill=black] (1.06,0.34) circle (2.5pt);
	\draw[color=black] (0.9,0.15) node {$z$};
	\end{scriptsize}
	\end{tikzpicture}
	\caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\end{figure}

Consider the unicyclic graph $G$ as shown in Figure \ref{fig1}. It is easy to see that $\gamma(G)=3$. We list all the $\gamma$-sets with properties which are among the five commonly studied, as  previously mentioned.

\begin{itemize}

	\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
	\item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
	 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
	\item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are  $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\displaystyle \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

A {\it dominating set} for a graph $G = (V,E)$ is a subset $S \subseteq V$ such that every vertex $v\in V$ is either in $S$ or has a neighbor $u\in S$. The vertex $u \in S$ is said to \textit{cover} the vertex $v \in V$ if either $u=v$ or $uv\in E$. A dominating set $S$ is a \textit{minimal dominating set} if no proper subset $S'\subset S$ is a dominating set. The \textit{domination number} $\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \textbf{$\gamma$-sets}.

\begin{enumerate}
 \item \textit{Perfect $\gamma$-set} requires that no two vertices in a $\gamma$-set cover the same vertex (outside the $\gamma$-set). \item \textit{Connected $\gamma$-set} requires that the graph induced by the $\gamma$-set be connected.
 \item \textit{Total $\gamma$-set} forbids isolated vertices in the subgraph induced by the $\gamma$-set.
 \item \textit{Independent $\gamma$-set} requires all vertices in a $\gamma$-set to be isolated vertices in the subgraph they induce.
 \item \textit{clique $\gamma$ set} requires that the vertices in a $\gamma$-set induce a clique.
\end{enumerate}

The \textit{dominion (number)} of a graph $G=(V,E)$, denoted by $\zeta(G)$, is the number of its $\gamma$-sets. In other words,

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
 \centering
 \begin{tikzpicture}[scale=1]
 \clip(0,-1) rectangle (4,4);
 \draw [line width=2pt] (2.62,1)-- (2,2);
 \draw [line width=2pt] (2,2)-- (1.98,2.84);
 \draw [line width=2pt] (2.62,1)-- (3,0.34);
 \draw [line width=2pt] (2,2)-- (1.48,1.02);
 \draw [line width=2pt] (1.48,1.02)-- (2.62,1);
 \draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
 \begin{scriptsize}
 \draw [fill=black] (2,2) circle (2.5pt);
 \draw[color=black] (2.2,2.1) node {$u$};
 \draw [fill=black] (2.62,1) circle (2.5pt);
 \draw[color=black] (2.92,1) node {$v$};
 \draw [fill=black] (1.98,2.84) circle (2.5pt);
 \draw[color=black] (1.94,3.1) node {$x$};
 \draw [fill=black] (3,0.34) circle (2.5pt);
 \draw[color=black] (3.2,0.15) node {$y$};
 \draw [fill=black] (1.48,1.02) circle (2.5pt);
 \draw[color=black] (1.14,1) node {$w$};
 \draw [fill=black] (1.06,0.34) circle (2.5pt);
 \draw[color=black] (0.9,0.15) node {$z$};
 \end{scriptsize}
 \end{tikzpicture}
 \caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\begin{itemize}

\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
 \item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
 \item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\displaystyle \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

\label{fig1}

\end{figure}

Consider the unicyclic graph $G$ as shown in Figure \ref{fig1}. It is easy to see that $\gamma(G)=3$. We list all the $\gamma$-sets with properties which are among the five

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
 \centering
 \begin{tikzpicture}[scale=1]
 \clip(0,-1) rectangle (4,4);
 \draw [line width=2pt] (2.62,1)-- (2,2);
 \draw [line width=2pt] (2,2)-- (1.98,2.84);
 \draw [line width=2pt] (2.62,1)-- (3,0.34);
 \draw [line width=2pt] (2,2)-- (1.48,1.02);
 \draw [line width=2pt] (1.48,1.02)-- (2.62,1);
 \draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
 \begin{scriptsize}
 \draw [fill=black] (2,2) circle (2.5pt);
 \draw[color=black] (2.2,2.1) node {$u$};
 \draw [fill=black] (2.62,1) circle (2.5pt);
 \draw[color=black] (2.92,1) node {$v$};
 \draw [fill=black] (1.98,2.84) circle (2.5pt);
 \draw[color=black] (1.94,3.1) node {$x$};
 \draw [fill=black] (3,0.34) circle (2.5pt);
 \draw[color=black] (3.2,0.15) node {$y$};
 \draw [fill=black] (1.48,1.02) circle (2.5pt);
 \draw[color=black] (1.14,1) node {$w$};
 \draw [fill=black] (1.06,0.34) circle (2.5pt);
 \draw[color=black] (0.9,0.15) node {$z$};
 \end{scriptsize}
 \end{tikzpicture}
 \caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
 \item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
 \item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\ds \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

\begin{proof}
For each vertex on the cycle, and for each dominating set $S\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives $\ds \sum_{i=0}^n{n\choose i}$. Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$. This gives $\zeta(G)=2^n$.
\end{proof}

\begin{claim}{\label{c4}}
 When $n=3k, k\ge 1$, $\zeta(P_n)=1$ and the unique $\gamma$-set in $P_n$ contains $v_{n-1}$. Moreover, when $n=3k+2, k\ge 1$, there is only one $\gamma$-set in $P_n$ containing $v_{n}$.
\end{claim}
\begin{proof}
 When $n=3k, k\ge 1$, we have $\zeta(P_n)=k$ and the conclusion follows by the pigeon-hold argument alluded in Claim \ref{c1}. Each of the sets of $3$ consecutive vertices $\{v_{3r+1},v_{3r+2},v_{3r+3}\}$, $r=0,\ldots,k-1$, must contain at least one vertex of any $\gamma$-set $S$, and since $\gamma=k$, each of those sets must contain exactly one element of $S$. Since $S$ is dominating, the element of $S$ in $\{v_1,v_2,v_3\}$ must be $v_1$ or $v_2$. If $v_1$ then the ``next'' element of $S$ must be $v_4$, which forces $S=\{v_1,v_4,\ldots,v_{3k-2}\}$, leaving $v_{3k}$ uncovered. Therefore, the first element of $S$ must be $v_2$, which forces $S=\{v_2,v_5,\ldots,v_{3k-1}, v_{3k+2}\}$.

\begin{proposition}{\label{Prop2}}
 For any connected graph $G$ on $n$ vertices, $1\le \zeta(G)\le {n\choose \gamma(G)}$.

\end{proof}
\begin{corollary}
For any connected graph $G$ with $\gamma(G)=1$, $\zeta(\underbrace{G\vee \ldots \vee G}_r)=r\zeta(G)$, $r\ge 1$.
\end{corollary}
\begin{proof}
From Part (a) of Theorem \ref{T4} (when $\gamma(G_1)=1=\gamma(G_2)$), the result follows by induction on $r\ge 1$.
\end{proof}
We note that in the case when $G=K_1$, it is clear that $\underbrace{G\vee \ldots \vee G}_r:=K_r$ and $\zeta(K_r)=r=r\zeta(K_1)$. Here, we present a more general result for the special case when $G=\overline{K_{m}}$, the complement of a complete graph on $m\ge 1$ vertices. Note that $\overline{K_{m_1}} \vee \ldots \vee \overline{K_{m_k}}$ is a complete $k$-partite graph often written as $K(m_1,m_2,\ldots, m_k)$, with $m_i\ge 1$ and $k\ge 2$.

\end{theorem}
\begin{proof}
Let $V_i$ denote each disjoint partite (vertex) set of $G$, with $|V_i|=m_i$, $i=1,\ldots, k$. If, for some $j=1,\ldots, r$, $m_j=1$ and $m_i>1$ for $i>r$, then it is clear that $\gamma(G)=1$ and the $\gamma$-sets in $G$ are the singleton subsets of $V_1\cup\ldots \cup V_r$. It follows that $\zeta(G)=r$.
Assume $m_i>1$, for $i=1,\ldots, k$. It is clear that $\gamma(G)=2$. Any vertex $v\in V_i$ covers all vertices $v'\in V(G)\setminus V_i$ except those in $V_i$. Thus, for each $v\in V_i$ and $v'\in V_j$, $i\ne j$, $\{v,v'\}$ is a $\gamma$-set of $G$. It follows that there are $\ds \sum_{i\ne j}^{{k}}m_im_j$ total such sets. Further, if $|V_i|=2$ for $i=1,\ldots, r$, then each such sets, $V_i$, is an additional $\gamma$-set, giving the result.
\end{proof}

\begin{proof}
For each vertex on the cycle, and for each dominating set $S\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives  $\displaystyle \sum_{i=0}^n{n\choose i}$.  Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$. This gives $\zeta(G)=2^n$.
\end{proof}

For each vertex on the cycle and each dominating set $S\subseteq V(G)$, "either the vertex or its leaf neighbor must be in $S$." Hence all $\gamma$-sets can be formed by choosing a subset of the $n$ leaves, and then including "the cycle neighbors of unchosen leafs," yielding $\sum_{i=0}^n \binom{n}{i}$. Conversely, "every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$," so $\zeta(G)=\sum_{i=0}^n \binom{n}{i}=2^n$.

If $G$ is a sun graph on $2n\ge 6$ vertices, then $\displaystyle \zeta(G)=\sum_{i=0}^n{n\choose i}=2^n. $,

Question: Let b3(G) denote the domination number of a graph G, i.e. the minimum size of a dominating set, and let b6(G) be the number of b3-sets of G, where a b3-set is a dominating set of cardinality b3(G). A sun graph on 2n vertices (with 2n \ge 6) is the graph obtained from a cycle C_n by attaching exactly one pendant leaf to each cycle vertex. Which quantitative statement holds for b6(G)?

If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\sum_{i=0}^{n}\binom{n}{i}=2^n\).

Choices:
B. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\binom{2n}{n}\).
C. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)\ge 2^n\).
D. If G is a sun graph on 2n \ge 6 vertices, then every \(\gamma\)-set is obtained by choosing exactly one vertex from each leaf-cycle pair, and hence \(\zeta(G)=n\).
E. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\sum_{i=1}^{n}\binom{n}{i}=2^n-1\).

independent binary choices at n cycle-leaf pairs replaced by unrestricted n-subset count from 2n vertices

replaced exact enumeration by the weaker lower bound only

local one-from-each-pair structure kept but global count collapsed from 2 choices per pair to one canonical choice

excluded one valid endpoint case in the binomial sum

The stem defines the sun graph and the counting function zeta(G), but it does not explicitly state or trivially reveal the value 2^n. The correct answer must still be derived from the graph structure.

This is not a mere restatement of a definition from the stem. The solver must determine gamma(G) and then characterize all minimum dominating sets, so the conclusion is not tautological.

Some genuine reasoning is required: one must see that every leaf-cycle pair forces at least one selected vertex, conclude gamma(G)=n, and then count the binary choices. However, once that structure is noticed, the answer becomes fairly direct rather than deeply generative.

Several distractors reflect plausible counting mistakes (off-by-one, overcounting, confusing all subsets of a given size with minimum dominating sets). However, choice C is actually true as a weaker statement, which makes the option set ambiguous in a single-answer MCQ and reduces distractor quality.

A mathematically meaningful item with no answer leakage and a non-tautological setup, but its discriminative quality is weakened by the presence of a weaker true distractor, making the single-correct format imperfect.