[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


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