Some Consequences of the Valley Delta Conjectures

Abstract

Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018) introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $$\Delta '_{e_{n-k-1}} e_n$$ Δ e n - k - 1 ′ e n in terms of rise-decorated or valley-decorated labelled Dyck paths. While the rise version has been recently proved (D’Adderio and Mellit in Adv Math 402:108342, 2022; Blasiak et al. in A Proof of the Extended Delta Conjecture, arXiv:2102.08815, 2021), not much is known about the valley version. In this work, we prove the Schröder case of the valley Delta conjecture, the Schröder case of its square version (Iraci and Wyngaerd in Ann Combin 25(1):195–227, 2021), and the Catalan case of its extended version (Qiu and Wilson in J Combin Theory Ser A 175:105271, 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 and Wyngaerd 2021).