The Schröder case of the generalized Delta conjecture

Abstract

We prove the Schröder case, i.e. the case $⟨\cdot ,e_{n-d}{h}_d⟩$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for ${\Delta }_{h_m}{\Delta }_{e_{n-k-1}}^{\prime }{e}_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schröder of Haglund (Haglund 2004). 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 (D'Adderio, Iraci 2017), which in turn extended results in (Aval et al. 2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations.

22 pages, 12 figures

References in corpus (4)

• A proof of the Delta Conjecture when $q=0$
• Decorated Dyck paths, the Delta conjecture, and a new q,t-square
• Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers
• Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture (FULL VERSION)

Cited by in corpus (8)

• Decorated Dyck paths, polyominoes, and the Delta conjecture
• The Delta square conjecture
• The generalized Delta conjecture at t=0
• A valley version of the Delta square conjecture
• Tiered trees and Theta operators
• Theta operators, refined Delta conjectures, and coinvariants
• Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors
• Some consequences of the valley Delta conjectures