Decorated square paths at q=-1

TL;DR

This paper investigates the evaluation at q=-1 of a symmetric function enumerator related to decorated square paths, revealing connections to Euler numbers and Dyck path enumerators, and introduces a cyclic group action called cutting and pasting.

Contribution

It provides a new combinatorial interpretation of the q=-1 evaluation of the symmetric function, using a cyclic group action on decorated paths, and links it to Euler numbers and Dyck path enumeration.

Findings

The q=-1 evaluation is zero when n-k is even.

When n-k is odd, the evaluation yields a positive polynomial related to Euler numbers.

The combinatorics connects decorated square paths with Dyck path enumerators.

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.