A basis of the alternating diagonal coinvariants

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.