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Two $t$-analogues of the tree inversion enumerator

· 2025 · math.CO · arXiv 2510.22385

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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.

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2026 1

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  • Short Proofs in Algebraic and Enumerative Combinatorics math.CO · 2026-05-19 · unverdicted · none · ref 19 · 3 links · internal anchor

    AI-generated short proofs resolve conjectures on the echelonmotion operator, parking function statistics, and plactic monoid centralizers.