Conjectured Statistics for the Higher $q,t$-Catalan Sequences

Abstract

This article describes conjectured combinatorial interpretations for the higher $q,t$-Catalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund and Haiman for the original $q,t$-Catalan sequence. We prove explicit summation formulas, bijections, and recursions involving the new statistics. We show that specializations of the combinatorial sequences obtained by setting $t=1$ or $q=1$ or $t=1/q$ agree with the corresponding specializations of the Garsia-Haiman sequences. A third statistic occurs naturally in the combinatorial setting, leading to the introduction of $q,t,r$-Catalan sequences. Similar combinatorial results are proved for these trivariate sequences.