Rational Parking Functions and Catalan Numbers

Drew Armstrong, Nicholas A. Loehr, Gregory S. Warrington

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


The “classical” parking functions, counted by the Cayley number (n+1)n−1, carry a natural permutation representation of the symmetric group Sn 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 ba−1, carry a permutation representation of Sa in which the number of orbits is the “rational” Catalan number (Formula presented.). First, we compute the Frobenius characteristic of the Sa-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 languageEnglish (US)
Pages (from-to)21-58
Number of pages38
JournalAnnals of Combinatorics
Issue number1
StatePublished - Mar 1 2016


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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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