Decorated square paths at q=-1

The valley Delta square conjecture states that the symmetric function [n-k]q/[n]qΔ_eₙ₋ₖω(pₙ) 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 . [n-k]q/[n]qΔ_eₙ₋ₖω(pₙ), 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_eₙ₋ₖ₋₁'eₙ,h₁ⁿ considered by Corteel-Josuat Verg\`{e}s-Vanden Wyngaerd.