Séminaire Lotharingien de Combinatoire, 93B.57 (2025), 12 pp.

Sylvie Corteel, Alexander Lazar and Anna Vanden Wyngaerd

Positivity Phenomena for Lattice Paths at q=-1

Abstract. The valley delta square conjecture proposes that the coefficients of the symmetric function [n-k]q- [n]qΔen-kω(pn) can be expressed in terms of a certain class of decorated square paths with respect to the bistatistic (dinv,area). Inspired by recent positivity results of Corteel, Josuat-Vergès, and Vanden Wyngaerd, we study the evaluation of this enumerator at q=-1. By considering a cyclic group action on the decorated square paths, we show that ⟨[n--k]q n⟩|| [n]q Δen-kω(pn),h1 | 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 ⟨Δ'en-k-1en,h1n⟩ considered by Corteel-Josuat Vergès-Vanden Wyngaerd.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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