Combinatorics of [formula omitted]-parking functions

Abstract

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 CH n ( 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 q n ( n − 1 ) / 2 CH n ( 1 / q , q ) = [ n + 1 ] q n − 1 .