Coding Parking Functions by Pairs of Permutations - Burman, Yurii; Shapiro, Michael (The Electronic Journal of Combinatorics, 2003)
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 n7 we describe in terms of admissible pairs the dimensions of the bi-graded components h_u,v of diagonal harmonics C [x_1,…,x_n;y_1,…,y_n]/S_n, i.e., polynomials in two groups of n variables modulo the diagonal action of symmetric group S_n.