The Electronic Journal of Combinatorics· 2026Q1
On a $q$-Analogue of the Zeta Polynomial of Posets
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- Q1SCImago
- 2026year
Short summary
A new $q$-analogue of the Zeta polynomial for finite posets is introduced, with coefficients depending on $q$.
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Key points
- Introduced a $q$-analogue of the Zeta polynomial for finite posets.
- Proved properties related to duality, product, and disjoint union.
- The leading term is a $q$-analogue of the number of maximal chains.
- The value at $q=0$ is essentially the characteristic polynomial.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. The Electronic Journal of Combinatorics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics