ON q-COUNTING OF NONCROSSING CHAINS AND PARKING FUNCTIONS

Yen Jen Cheng, Sen Peng Eu, Tung Shan Fu, Jyun Cheng Yao

Research output: Contribution to journalArticlepeer-review

Abstract

For a finite Coxeter group W, Josuat-Verg\`es derived a q-polynomial counting the maximal chains in the lattice of noncrossing partitions of W by weighting some of the covering relations, which we call bad edges, in these chains with a parameter q. We study the connection of these weighted chains with parking functions of type A (B, respectively) from the perspective of the q-polynomial. The q-polynomial turns out to be the generating function for parking functions (of either type) with respect to the number of cars that do not park in their preferred spaces. In either case, we present a bijective result that carries bad edges to unlucky cars while preserving their relative order. Using this, we give an interpretation of the \gamma-positivity of the q-polynomial in the case when W is the hyperoctahedral group.

Original languageEnglish
Pages (from-to)917-946
Number of pages30
JournalSIAM Journal on Discrete Mathematics
Volume38
Issue number1
DOIs
Publication statusPublished - 2024

Keywords

  • absolute order
  • Coxeter group
  • noncrossing partition
  • parking functions

ASJC Scopus subject areas

  • General Mathematics

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