Coding parking functions by pairs of permutations. (English) Zbl 1023.05007

Burman, Yurii; Shapiro, Michael

Electron. J. Comb. 10, Research paper R23, 8 p. (2003); printed version J. Comb. 10, No. 3 (2003).

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 \(S_n\).

MSC:

05A15	Exact enumeration problems, generating functions
05A19	Combinatorial identities, bijective combinatorics
16S36	Ordinary and skew polynomial rings and semigroup rings