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.

### The Delta Square Conjecture

#### Abstract

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.
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2021
D'Adderio, M; Iraci, Alessandro; Anna Vanden Wyngaerd,
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11568/1119672`