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Peer-reviewedCC BY 4.0The Electronic Journal of Combinatorics

On a q-Analogue of the Zeta Polynomial of Posets

  • Frédéric Chapoton
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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.

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Cite

BibTeXRIS
BibTeX
@article{chapoton2026qanalogue,
  title = {{On a $q$-Analogue of the Zeta Polynomial of Posets}},
  author = {Frédéric Chapoton},
  journal = {The Electronic Journal of Combinatorics},
  year = {2026},
  month = oct,
  doi = {10.37236/13820},
  url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p21},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On a q-Analogue of the Zeta Polynomial of Posets
AU  - Frédéric Chapoton
PY  - 2026
DA  - 2026/10/23
JO  - The Electronic Journal of Combinatorics
DO  - 10.37236/13820
UR  - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p21
AB  - 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.
SN  - 1077-8926
ER  -