The Schröder case of the generalized Delta conjecture

Abstract

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.

The Delta Square Conjecture

“…the case ·, e n−d h d of our generalized Delta square conjecture. This is the analogue of the same result for the generalized Delta conjecture proved in [6], and it is a broad generalization of the q, t-square theorem proved in [3]. In Section 8 we give a combinatorial involution that will provide a counterpart of two theorems on symmetric functions proved in Section 5.…”

“…Also, we prove the Schröder case, i.e. the case ·, e n−d h d , of the generalized Delta square conjecture: this is a broad generalization of the q, t-square theorem of Can and Loehr [3], and it is the analogue of the same result for the generalized Delta conjecture proved in [6]. Finally, we provide a combinatorial involution among the objects of the generalized Delta (square) conjectures for fixed m and n. Together with its symmetric function counterpart and the specialization q = 0 of the generalized Delta conjecture at k = 0, this will prove a curious linear relation among such conjectures.…”

“…With regard to this labelling the Definitions 7.4 and 3.9 of the dinv coincide. * For example, the path in Figure 4 is the partially labelled square path corresponding to the decorated square path in Figure 3 We recall here the main result from [6]. n,k;p .…”

“…The following definitions extend the ones in [6,Section 4] from Dyck paths to square paths (ending east). Definition 7.1.…”

The New dinv is Not So New

“…In fact we show how the ndinv is none other than the natural dinv statistic, but read on the appropriate subset of partially labelled Dyck paths. We do this by combining two bijections in [4] and [3] that involve parallelogram polyominoes.…”