Combinatorics of the Delta conjecture at q=-1

TL;DR

This paper explores the combinatorial structure of the Delta conjecture at q=-1, revealing connections to Euler numbers and permutation statistics through symmetric functions and parking functions.

Contribution

It establishes a new link between the Delta operator at q=-1 and Euler numbers, introducing novel permutation statistics and extending the combinatorial understanding of the conjecture.

Findings

At q=-1, the q,t-analog relates to Euler numbers.

New permutation statistics involving peaks and valleys are introduced.

Empirical evidence suggests broader nonnegativity phenomena.

Abstract

In the context of the shuffle theorem, many classical integer sequences appear with a natural refinement by two statistics $q$ and $t$: for example the Catalan and Schr\"oder numbers. In particular, the bigraded Hilbert series of diagonal harmonics is a $q,t$-analog of $(n+1)^{n-1}$ (and can be written in terms of symmetric functions via the nabla operator). The motivation for this work is the observation that at $q=-1$, this $q,t$-analog becomes a $t$-analog of Euler numbers, a famous integer sequence that counts alternating permutations. We prove this observation via a more general statement, that involves the Delta operator on symmetric functions (on one side), and new combinatorial statistics on permutations involving peaks and valleys (on the other side). An important tool are the schedule numbers of a parking function first introduced by Hicks; and expanded upon by Haglund and Sergel. Other empirical observation suggest that nonnegativity at $q=-1$ holds in far greater generality.