On a $q$-Analogue of the Zeta Polynomial of Posets

Abstract

We introduce a $q$-analogue of the classical Zeta polynomial of fiinite partially ordered sets, as a polynomial in one variable $x$ with coefficients depending on the indeterminate $q$. We prove some properties of this polynomial invariant, including its behaviour with respect to duality, product and disjoint union. The leading term is a $q$-analogue of the number of maximal chains, but not always with non-negative coefficients. The value at $q=0$ turns out to be essentially the characteristic polynomial.