### 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 Catalan number (Formula presented.). In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by b^{a−1}, carry a permutation representation of S_{a} in which the number of orbits is the “rational” Catalan number (Formula presented.). First, we compute the Frobenius characteristic of the S_{a}-module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers (Formula presented.) and for the q-binomial coefficients (Formula presented.). We give a bijective explanation of the division by [a+b]_{q} that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.

Original language | English (US) |
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Pages (from-to) | 21-58 |

Number of pages | 38 |

Journal | Annals of Combinatorics |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2016 |

### Keywords

- diagonal harmonics
- q, t-Catalan numbers
- rational Catalan numbers
- rational parking functions
- Shuffle Conjecture

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

*Annals of Combinatorics*,

*20*(1), 21-58. https://doi.org/10.1007/s00026-015-0293-6