A new recursion for three-column combinatorial Macdonald polynomials

The Hilbert series F̃μ of the Garsia-Haiman module Mμ can be described combinatorially as the generating function of certain fillings of the Ferrers diagram of μ where μ is an integer partition of n. Since there are n! fillings that generate F̃μ, it is desirable to find recursions to reduce the number of fillings that need to be considered when computing F̃μ combinatorially. In this paper, we present a combinatorial recursion for the case where μ is an n by 3 rectangle. This allows us to reduce the number of fillings under consideration from (3n)! to (3n)!/(3!n!).