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On a q-analogue of the Zeta polynomial of posets

Frédéric Chapoton

TL;DR

The paper defines a $q$-analogue of the Zeta polynomial for finite posets equipped with a height function, connecting it to $q$-Ehrhart polynomials and $P$-partition theory. It establishes fundamental properties like multiplicativity under products, additivity under disjoint unions, and key duality formulas, while linking to the $q$-order polynomial in distributive lattices. It develops a rich Ehrhart-theoretic framework, including $q$-Ehrhart series and $q$-volumes, and identifies positivity criteria via $R$-labelings, with applications to Coxeter-related posets. A surprising bridge to classical invariants emerges at $q=0$, where the specialization relates to the characteristic polynomial, and an Eulerian reciprocity theorem extends to the $q$-setting. Overall, the work integrates $q$-Ehrhart theory with poset invariants, yielding a coherent platform for $q$-analogues of zeta-type polynomials and their combinatorial-indexed generating series.

Abstract

We introduce a q-analogue of the classical Zeta polynomial of finite 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.

On a q-analogue of the Zeta polynomial of posets

TL;DR

The paper defines a -analogue of the Zeta polynomial for finite posets equipped with a height function, connecting it to -Ehrhart polynomials and -partition theory. It establishes fundamental properties like multiplicativity under products, additivity under disjoint unions, and key duality formulas, while linking to the -order polynomial in distributive lattices. It develops a rich Ehrhart-theoretic framework, including -Ehrhart series and -volumes, and identifies positivity criteria via -labelings, with applications to Coxeter-related posets. A surprising bridge to classical invariants emerges at , where the specialization relates to the characteristic polynomial, and an Eulerian reciprocity theorem extends to the -setting. Overall, the work integrates -Ehrhart theory with poset invariants, yielding a coherent platform for -analogues of zeta-type polynomials and their combinatorial-indexed generating series.

Abstract

We introduce a q-analogue of the classical Zeta polynomial of finite 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.
Paper Structure (17 sections, 30 theorems, 80 equations)

This paper contains 17 sections, 30 theorems, 80 equations.

Table of Contents

  1. $q$-analogue
  2. $q$-Ehrhart polynomials and reciprocity
  3. Definition of $q$-Zeta polynomial
  4. Examples
  5. Basic properties
  6. Relation with the $q$-order polynomial
  7. Ehrhart series and volumes
  8. $P$-partitions and $q$-Ehrhart series for posets
  9. $q$-Ehrhart series for $q$-Zeta polynomials
  10. Positivity properties
  11. $R$-labellings
  12. Positivity criterion
  13. Value at $q=0$ and characteristic polynomial
  14. Reciprocity for Eulerian posets
  15. Flag $f$-vectors and $h$-vectors
  16. ...and 2 more sections

Key Result

Lemma 1.1

The values of $\mathsf{Z}_{P,h}$ at $q$-integers $[n]_q$ for $n \geq 2$ are $q$-analogues of the numbers of chains $e_1 \leq \dots \leq e_{n-1}$ in $P$, where the power of $q$ is the sum of the heights of elements in the chain.

Theorems & Definitions (67)

  • Definition 1
  • Lemma 1.1
  • Lemma 1.2
  • proof
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Example 1.8
  • ...and 57 more