Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture

TL;DR

This paper introduces new combinatorial statistics on parallelogram polyominoes and Dyck paths, establishing connections to the Delta conjecture and providing new interpretations for related symmetric function coefficients.

Contribution

It extends previous work by defining new statistics and bijections that link parallelogram polyominoes and Dyck paths to key conjectures in algebraic combinatorics.

Findings

Matching q,t-enumerators with symmetric function coefficients

Bijective connection between polyominoes and Dyck paths

New conjectural combinatorial interpretation of Delta operator expressions

Abstract

We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match $\langle \Delta_{h_m} e_{n+1}, s_{k+1,1^{n-k}} \rangle$, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient $\langle\Delta_{e_{m+n-k-1}}'e_{m+n}, h_m h_n \rangle$ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially labelled Dyck paths, which provides another conjectural combinatorial interpretation of $\Delta_{h_{\ell}}\Delta_{e_{n-k-1}}'e_n$, cf. (Haglund et al. 2015). This is an extended abstract of (D'Adderio, Iraci 2017): this forthcoming publication will have proofs and additional details and results.