Rational Parking Functions and Catalan Numbers

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 1/n+1( [ 2n; n ]) . 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 1/a+b( [ a+b; a ]) . 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 1/[a+b]_q[ [ a+b; a ]]_q and for the q -binomial coefficients [ [ n; k ]]_q . 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 .