The Delta Square Conjecture

Abstract

We conjecture a formula for the symmetric function [n−k]t [n]t ∆ hm ∆e n−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 [20], recently proved by Sergel [25] 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 [14], 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 Schröder case, i.e. the case ·, e n−d h d . 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.Partially labelled Dyck paths differ from labelled Dyck paths only in that 0 is allowed as a label in the former and not in the latter.Definition 2.2. We define for each D ∈ PLD(m, n) a monomial in the variables x 1 , x 2 , . . . : we setwhere l i (D) is the label of the i-th vertical step of D (the first being at the bottom). Notice that x 0 does not appear, which explains the word partially.

The generalized Delta conjecture at t=0

“…Our theorem generalizes recent results by Garsia, Haglund, Remmel and Yoo [6]. This proves also the case q = 0 of our recent generalized Delta square conjecture [3].…”

“…As it is known that PLD x,q,t (m, n) * n−k is a symmetric function (see for example a similar argument sketched in [3,Remark 3.13]), this shows that OPd R x,q (m, n) k is a symmetric function as well.…”

“…Finally, in [3] we proposed what we called a generalized Delta square conjecture, which extends the square conjecture of Loehr and Warrington [14], recently proved by Sergel [16], and it reduces to the generalized Delta conjecture when q = 0: hence we proved also the case q = 0 of this newer conjecture.…”

“…Remark 1.2. We would like to emphasize that our proof is independent of the results in [6], hence providing a further new proof of the Delta conjecture at q = 0 or t = 0, after the alternative proofs in [11] and in [3]. It should also be noticed that our proof has the peculiar property that it does not specialize to the case m = 0: for our argument to go through, the full generalized Delta conjecture at t = 0 or q = 0 is needed.…”

“…The argument to prove this is the same as the one appearing in the proof of Theorem 5.1 in [3] in the case m = 0.…”