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arXiv · 1403.1845

Rational parking functions and Catalan numbers

Drew Armstrong·Nicholas A. Loehr·Gregory S. Warrington
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Abstract

The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.

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Drew Armstrong, Nicholas A. Loehr, Gregory S. Warrington. 2014-03-07. Rational parking functions and Catalan numbers. https://arxiv.org/abs/1403.1845

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