The zeta polynomial 
of a nonempty finite partially
ordered set 
is the polynomial whose value at each integer 
counts weakly increasing sequences
of 
elements of 
:
 |
If 
is the number of strict chains with 
elements, then
 |
Only finitely many terms are nonzero. This expression shows that 
has coefficients that are
rational numbers and takes integer
values at integers (Stanley 2012, pp. 291-292; Chapoton
2026, p. 3).
For an antichain of 
elements, 
. For a chain of 
elements, 
. The zeta polynomial is multiplicative under
the Cartesian product of partially
ordered sets and additive under their disjoint union (Chapoton 2026, p. 3).
Its height-weighted q-analog is the q-zeta
polynomial.
-Analogue of the Zeta Polynomial of Posets." Elec.
J. Combin. 33, P4.21, 2026.