No abstract

Inspired by [Qiu, Wilson 2020] and [D'Adderio, Iraci, Vanden Wyngaerd 2019], we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in [Haglund, Sergel 2019] to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in [Sergel 2016], relying on the formulation of the touching version in terms of the Θ f operators introduced in [D'Adderio, Iraci, Vanden Wyngaerd 2020-Theta Operators].

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.

We prove the Schröder case, i.e. the case ·, e n−d h d , of the conjecture of Haglund Remmel and Wilson [11] for ∆ hm ∆ ′ e n−k−1 en in terms of decorated partially labelled Dyck paths, which we call generalized Delta conjecture. This result extends the Schröder case of the Delta conjecture proved in [5], which in turn generalized the q, t-Schröder of Haglund [8]. The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in [4], which in turn extended results in [1]. Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations.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.Definition 2.3. Let D be a (partially labelled) Dyck path of size n + m. We define its area word to be the string of integers a(D) = a 1 (D) · · · a n+m (D) where a i (D) is the number of whole squares in the i-th row (from the bottom) between the path and the main diagonal. Definition 2.4. The rises of a Dyck path D are the indices Rise(D) := {2 ≤ i ≤ n + m | a i (D) > a i−1 (D)},or the vertical steps that are directly preceded by another vertical step. Taking a subset DRise(D) ⊆ Rise(D) and decorating the corresponding vertical steps with a * , we obtain a decorated Dyck path, and we will refer to these vertical steps as decorated rises.