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Two $t$-analogues of the tree inversion enumerator

Sam Hopkins

Abstract

In this note, we introduce two $t$-analogues $I_n(q,t)$ and $\widetilde{I}_n(q,t)$ of the tree inversion enumerator $I_n(q)$. Although similar, $I_n(q,t)$ and $\widetilde{I}_n(q,t)$ are different. But they both seem to have interesting properties. In particular, we conjecture that their $q=-1$ specializations give two different, natural refinements of the zigzag numbers counting alternating permutations.

Two $t$-analogues of the tree inversion enumerator

Abstract

In this note, we introduce two -analogues and of the tree inversion enumerator . Although similar, and are different. But they both seem to have interesting properties. In particular, we conjecture that their specializations give two different, natural refinements of the zigzag numbers counting alternating permutations.
Paper Structure (1 section, 2 theorems, 9 equations, 2 tables)

This paper contains 1 section, 2 theorems, 9 equations, 2 tables.

Table of Contents

  1. Acknowledgments

Key Result

Theorem 1

We have and

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2: Stanley--Yin stanley2023enumerative
  • Conjecture 3
  • Remark 4
  • Conjecture 5