A basis of the alternating diagonal coinvariants

TL;DR

This paper constructs an explicit basis for the alternating part of the diagonal coinvariant ring using Vandermonde determinants, providing a combinatorial formula for the q,t-Catalan numbers and a new decomposition of m-Dyck paths.

Contribution

It introduces a new explicit basis for the diagonal coinvariant ring's alternating component and connects it to combinatorial formulas and path decompositions.

Findings

Explicit basis for the alternating component of the diagonal coinvariant ring.

Recovery of the combinatorial formula for the q,t-Catalan numbers.

A novel decomposition of m-Dyck paths into Dyck paths with combined area and bounce sequences.

Abstract

We construct an explicit vector space basis in terms of bivariate Vandermonde determinants for the alternating component of the diagonal coinvariant ring $DR_n$, answering a question of Stump. As a Corollary, we recover the combinatorial formula of the $q,t$-Catalan numbers. Moreover, we construct a decomposition of an $m$-Dyck path into an $m$-tuple of Dyck paths such that the area sequence and bounce sequence of the $m$-Dyck path is entrywise the sum of the area sequences and bounce sequences of the Dyck paths in the tuple.