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\begin{document}

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\section*{zeta polynomial}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{combinatorics}{}\paragraph*{{Combinatorics}}\label{combinatorics}

[[!include combinatorics - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_order_polynomial}{Relation to order polynomial}\dotfill \pageref*{relation_to_order_polynomial} \linebreak
\noindent\hyperlink{relation_to_zeta_function}{Relation to zeta function}\dotfill \pageref*{relation_to_zeta_function} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{zeta polynomial} $Z_P(n)$ of a finite [[partially ordered set]] $P$ counts the number of multichains (also known as ``weakly increasing sequences'') of length $n$ in $P$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

By a \emph{multichain} of length $n$ in $P$, we mean a sequence of elements $x_0 \le x_1 \le \cdots \le x_n$, which can be identified with an [[order-preserving function]] from the [[linear order]] $[n] = \{ 0 \lt 1 \lt \cdots \lt n \}$ into $P$. To see that

\begin{displaymath}
Z_P(n) = |Hom([n],P)|
\end{displaymath}
defines a [[polynomial]] in $n$,\footnote{Note that the definition we use here for $Z_P(n)$ has an index shift from the definition that seems to be more standard in combinatorics. For example, the definition in \hyperlink{StanleyEC1}{(Stanley, 3.12)} counts multichains of length $n-2$ rather than of length $n$. Accordingly, one should apply a substitution to get some of the properties stated here to match equivalent results in the literature.}  first observe that any function $[n] \to P$ factors as a [[surjection]] from $[n]$ onto some $[k] = \{ 0 \lt 1 \lt \cdots \lt k \}$ (where $k \le n$), followed by an [[injection]] from $[k]$ to $P$. The total number of order-preserving functions from $[n]$ to $P$ can therefore be calculated explicitly as

\begin{displaymath}
Z_P(n) = \sum_{k=0}^{d} b_k \binom{n}{k}
\end{displaymath}
where $b_k$ is the number of \emph{chains} $x_0 \lt x_1 \lt \cdots \lt x_k$ in $P$ (i.e., injective order-preserving functions from $[k]$ to $P$), and where $d$ is the length of the longest chain. Hence $Z_P(n)$ is a polynomial of degree equal to the length of the longest chain in $P$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The zeta polynomial of $[2] = \{ 0 \lt 1 \lt 2 \}$ is

\begin{displaymath}
3 + 3n + \binom{n}{2} = \frac{n^2 + 5n + 6}{2}
\end{displaymath}
For example, evaluating the polynomial at $n=0$ and $n=1$ confirms that $[2]$ contains 3 points and 6 [[intervals]], while evaluating it at $n=2$ confirms that there are 10 order-preserving functions from $[2]$ to itself.

The zeta polynomial of the 5-element poset

\begin{displaymath}
P = \itexarray{&&v&& \\ &&\uparrow&& \\ &&u&& \\ &\nearrow& &\nwarrow& \\ y &&&& z \\ &\nwarrow& &\nearrow& \\ &&x&&}
\end{displaymath}
is $5 + 9n + 7\binom{n}{2} + 2\binom{n}{3} = \frac{2n^3 + 15n^2 + 37n + 30}{6}$. Evaluating at $n=1$, we compute that $P$ contains 14 distinct intervals.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_order_polynomial}{}\subsubsection*{{Relation to order polynomial}}\label{relation_to_order_polynomial}

The [[order polynomial]] is related to the zeta polynomial by the equation

\begin{displaymath}
\Omega_P(n+2) = Z_{P^\downarrow}(n)
\end{displaymath}
where $P^\downarrow \cong (2)^P$ is the lattice of [[lower sets]] in $P$. This can be seen as a consequence of the [[currying]] isomorphisms

\begin{displaymath}
Hom([n], P^\downarrow) \cong Hom([n] \times P, (2)) \cong Hom(P, [n]^\downarrow)
\end{displaymath}
together with the isomorphisms $[n]^\downarrow \cong [n+1] \cong (n+2)$.

\hypertarget{relation_to_zeta_function}{}\subsubsection*{{Relation to zeta function}}\label{relation_to_zeta_function}

Using the formalism of [[incidence algebras]], the zeta polynomial has a simple expression in terms of the \emph{zeta function} of $P$ (defined by $\zeta_P(x,y) = 1$ if $x\le y$ and $\zeta_P(x,y) = 0$ otherwise):

\begin{displaymath}
Z_P(n) = \sum_{x,y\in P} \zeta_P^n(x,y)
\end{displaymath}
where $\zeta_P^n$ is the $n$-fold convolution product of $\zeta_P$. (In other words, if we view the zeta function as a square matrix, then the zeta polynomial is the sum of the entries in its $n$-fold matrix product.) This follows immediately from the definition of the convolution product,

\begin{displaymath}
(f\cdot g)(x,y) = \sum_{x \le z \le y} f(x,z) \cdot g(z,y)
\end{displaymath}
since $\zeta_P^n(x,y)$ computes the number of multichains of length $n$ in $P$ from $x$ to $y$.

As a special case, if $P$ has both a [[bottom]] element 0 and a [[top]] element 1, then

\begin{displaymath}
Z_P(n) = \zeta_P^{n+2}(0,1)
\end{displaymath}
since an arbitrary multichain $x_0 \le x_1 \le \cdots \le x_n$ of length $n$ can be extended to a multichain $0 \le x_0 \le x_1 \le \cdots \le x_n \le 1$ of length $n+2$ between 0 and 1.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[order polynomial]]
\item [[incidence algebra]]
\item [[Möbius inversion]]
\item [[nerve]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Paul H. Edeleman. Zeta Polynomials and the M\"o{}bius Function. European Journal of Combinatorics 1(4), 1980.

\end{itemize}
\begin{itemize}%
\item Richard P. Stanley, \emph{Enumerative combinatorics}, vol.1 (\href{http://www-math.mit.edu/~rstan/ec/ec1.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item Joseph P. S. Kung, [[Gian-Carlo Rota]], Catherine H. Yan. Combinatorics: The Rota Way. Cambridge, 2009.

\end{itemize}
category: combinatorics



\end{document}