Decorated square paths at q=-1

2024/08/20 by Sylvie Corteel, Corteel, Sylvie, Alexander Lazar +3
Computer Science · Engineering · #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2408.10640

openalex publication_date 2024/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The valley Delta square conjecture states that the symmetric function ([n-k]_q)/([n]_q)Δ_e_n-kω(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-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 which we call cutting and pasting, we show that .⟨ ([n-k]_q)/([n]_q)Δ_e_n-kω(p_n), h₁ⁿ⟩|_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_e_n-k-1'e_n,h₁ⁿ⟩ considered by Corteel-Josuat Vergès-Vanden Wyngaerd.

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