TY - JOUR

T1 - Rational Parking Functions and Catalan Numbers

AU - Armstrong, Drew

AU - Loehr, Nicholas A.

AU - Warrington, Gregory S.

N1 - Funding Information:
This work was partially supported by a grant from the Simons Foundation (#244398 to Nicholas Loehr). Author supported in part by National Science Foundation grant DMS-1201312.
Publisher Copyright:
© 2015, Springer International Publishing.

PY - 2016/3/1

Y1 - 2016/3/1

N2 - 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.

AB - 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.

KW - Shuffle Conjecture

KW - diagonal harmonics

KW - q, t-Catalan numbers

KW - rational Catalan numbers

KW - rational parking functions

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U2 - 10.1007/s00026-015-0293-6

DO - 10.1007/s00026-015-0293-6

M3 - Article

AN - SCOPUS:84958748120

VL - 20

SP - 21

EP - 58

JO - Annals of Combinatorics

JF - Annals of Combinatorics

SN - 0218-0006

IS - 1

ER -