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

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
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