Combinatorics of q,t-parking functions

In [A conjectured combinatorial formula for the Hilbert series for diagonal harmonics, in: Proceedings of FPSAC 2002 Conference, Melbourne, Australia, Discrete Math., in press] Haglund, Haiman, and the present author conjectured a combinatorial formula CHn(q,t) for the Hilbert series of diagonal harmonics as a weighted sum of parking functions. Another equivalent combinatorial formula was proposed by the present author in [Multivariate analogues of Catalan numbers, parking functions, and their extensions, UCSD doctoral thesis, June 2003]. These formulas involve three statistics on parking functions called area, dinv, and pmaj. In this article, we use the pmaj statistic to solve several combinatorial problems posed in [A conjectured combinatorial formula for the Hilbert series for diagonal harmonics, in: Proceedings of FPSAC 2002 Conference, Melbourne, Australia, Discrete Math., in press]. In particular, we derive a recursion satisfied by the combinatorial Hilbert series and show that qn(n−1)/2CHn(1/q,q)=[n+1]qn−1.