Two $t$-analogues of the tree inversion enumerator

TL;DR

This paper introduces two new $t$-analogues of the tree inversion enumerator, exploring their properties and conjecturing their connections to zigzag numbers and alternating permutations.

Contribution

The paper presents two novel $t$-analogues of the tree inversion enumerator and investigates their properties and potential combinatorial significance.

Findings

Both $t$-analogues are different but exhibit interesting properties.

Conjecture that their $q=-1$ specializations refine zigzag numbers.

Potential connections to counting alternating permutations.

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.