### Recursions and divisibility properties for combinatorial Macdonald polynomials

*Nicholas A. Loehr, Elizabeth M. Niese*

#### Abstract

For each integer partition µ, let
F

_{µ}(q,t) be the coefficient of x_{1}⋯x_{n}in the modified Macdonald polynomial H_{µ}. The polynomial F_{µ}(q,t) can be regarded as the Hilbert series of a certain doubly-graded S_{n}-module M_{µ}, or as a q,t-analogue of n! based on permutation statistics inv_{µ}and maj_{µ}that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of F_{µ}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 F_{µ}(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!2^{n}indexed by perfect matchings.Full Text: PDF PostScript