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

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.