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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. We conjecture that this decomposition gives a basis for the alternating component of the generalized diagonal coinvariants $DR_n^{(m)}$.