Some consequences of the valley Delta conjectures

TL;DR

This paper advances the understanding of the valley Delta conjecture by proving several of its specific cases, including Schr"oder and Catalan cases, and discusses implications assuming certain symmetries.

Contribution

It proves key cases of the valley Delta conjecture and its variants, which were previously unresolved, and explores implications under symmetry assumptions.

Findings

Proved the Schr"oder case of the valley Delta conjecture.

Proved the Schr"oder case of its square version.

Proved the Catalan case of its extended version.

Abstract

In (Haglund, Remmel, Wilson 2018) Haglund, Remmel and Wilson introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $\Delta'_{e_{n-k-1}} e_n$ in terms of rise-decorated or valley-decorated labelled Dyck paths respectively. While the rise version has been recently proved (D'Adderio, Mellit 2021; Blasiak, Haiman, Morse, Pun, Seelinger preprint 2021), not much is known about the valley version. In this work we prove the Schr\"oder case of the valley Delta conjecture, the Schr\"oder case of its square version (Iraci, Vanden Wyngaerd 2021), and the Catalan case of its extended version (Qiu, Wilson 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci, Vanden Wyngaerd 2021).