Recursions and Divisibility Properties for Combinatorial Macdonald Polynomials
For each integer partition µ, let Fµ(q,t) be the coefficient of x1⋯xn in the modified Macdonald polynomial Hµ. The polynomial Fµ(q,t) can be regarded as the Hilbert series of a certain doubly-graded Sn-module Mµ, or as a q,t-analogue of n! based on permutation statistics invµ and majµ that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of Fµ to prove some recursions characterizing these polynomials, and other related ones, when µ is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all µ, we show that Fµ(q,t) is divisible by certain q-factorials and t-factorials depending on µ. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express F(2n)(q,t) as a sum of q,t-analogues of n!2n indexed by perfect matchings.
Loehr, N. A., & Niese, E. M. (2011). Recursions and divisibility properties for combinatorial Macdonald polynomials. Discrete Mathematics & Theoretical Computer Science, 13(1), 21-44.