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arXiv · 2408.10640

Decorated square paths at q=-1

Sylvie Corteel·Alexander Lazar·Anna Vanden Wyngaerd
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Abstract

The valley Delta square conjecture states that the symmetric function $\frac{[n-k]_q}{[n]_q}\Delta_{e_{n-k}}\omega(p_n)$ can be expressed as the enumerator of a certain class of decorated square paths with respect to the bistatistic (dinv,area). Inspired by recent positivity results of Corteel, Josuat-Verg\`{e}s, and Vanden Wyngaerd, we study the evaluation of this enumerator at $q=-1$. By considering a cyclic group action on the decorated square paths which we call cutting and pasting, we show that $\left.\left\langle \frac{[n-k]_q}{[n]_q}\Delta_{e_{n-k}}\omega(p_n), h_1^n\right\rangle\right|_{q=-1}$ is $0$ whenever $n-k$ is even, and is a positive polynomial related to the Euler numbers when $n-k$ is odd. We also show that the combinatorics of this enumerator is closely connected to that of the Dyck path enumerator for $\langle\Delta_{e_{n-k-1}}'e_n,h_1^n\rangle$ considered by Corteel-Josuat Verg\`{e}s-Vanden Wyngaerd.

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Sylvie Corteel, Alexander Lazar, Anna Vanden Wyngaerd. 2024-08-20. Decorated square paths at q=-1. https://arxiv.org/abs/2408.10640

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