The Delta Square Conjecture

We conjecture a formula for the symmetric function [n.k]t/[n]t Δhm Δen-k ω(pn) in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington [23], recently proved by Sergel [28] after the breakthrough of Carlsson and Mellit [4]. Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel, and Wilson [17], answering one of their questions. We support our conjecture by proving the specialization m = q = 0, reducing it to the same case of the Delta conjecture, and the Schroder case, that is, the case (·, en-dhd). The latter provides a broad generalization of the q, t-square theorem of Can and Loehr [3]. We give also a combinatorial involution, which allows to establish a linear relation among our conjectures (as well as the generalized Delta conjectures) with fixed m and n. Finally, in the appendix, we give a new proof of the Delta conjecture at q = 0.