The 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 $P$.
By a 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
defines a polynomial in $n$,^{1} 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
where $b_k$ is the number of 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 $P$.
The zeta polynomial of $[2] = \{ 0 \lt 1 \lt 2 \}$ is
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
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.
The order polynomial is related to the zeta polynomial by the equation
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
together with the isomorphisms $[n]^\downarrow \cong [n+1] \cong (n+2)$.
Using the formalism of incidence algebras, the zeta polynomial has a simple expression in terms of the zeta function of $P$ (defined by $\zeta_P(x,y) = 1$ if $x\le y$ and $\zeta_P(x,y) = 0$ otherwise):
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,
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
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.
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 (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. ↩
Last revised on August 26, 2018 at 08:21:27. See the history of this page for a list of all contributions to it.