Deficit and $(q,t)$-symmetry in triangular partitions

TL;DR

This paper explores the $(q,t)$-enumeration of triangular Dyck paths, introducing new combinatorial objects and statistics to derive results and conjectures in the area.

Contribution

It introduces triangular and sim-sym tableaux and the deficit statistic, providing new insights and proofs for triangular partitions and a conjecture on lattice interval enumeration.

Findings

New interpretation of dinv via the deficit statistic

Results on triangular 2-partitions

A conjecture on lattice interval $(q,t,r)$-enumeration

Abstract

We study the $(q,t)$-enumeration of triangular Dyck paths considered by Bergeron and Mazin. To do so, we introduce the notion of triangular and sim-sym tableaux and the deficit statistic which is a new interpretation of the dinv. We use it to obtain new results and proofs on triangular $2$-partitions and an interesting conjecture for a certain lattice interval $(q,t,r)$-enumeration.