The Schr\"oder case of the generalized Delta conjecture

TL;DR

This paper proves the Schr"oder case of the generalized Delta conjecture using decorated Dyck paths, introduces new combinatorial interpretations, and extends previous results with recursive formulas and bijections.

Contribution

It establishes the Schr"oder case of the generalized Delta conjecture and provides multiple new combinatorial interpretations and recursive structures.

Findings

Proved the Schr"oder case of the generalized Delta conjecture.

Introduced a new bounce statistic for polynomial interpretation.

Extended combinatorial models to parallelogram polyominoes.

Abstract

We prove the Schr\"oder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schr\"oder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schr\"oder 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.