Coding parking functions by pairs of permutations (Q1408526)

Summary: We introduce a new class of admissible pairs of triangular sequences and prove a bijection between the set of admissible pairs of triangular sequences of length \(n\) and the set of parking functions of length \(n\). For all \(u\) and \(v=0,1,2,3\) and all \(n \leq 7\) we describe in terms of admissible pairs the dimensions of the bi-graded components \(h_{u,v}\) of diagonal harmonics \(\mathbb{C}[x_1,\dots,x_n;y_1,\dots,y_n]/S_n\), i.e., polynomials in two groups of \(n\) variables modulo the diagonal action of symmetric group \(S_n\).