A new recursion for three-column combinatorial Macdonald polynomials

摘要

The Hilbert series F~μ of the Garsia-Haiman module M _μ 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 F~μ, it is desirable to find recursions to reduce the number of fillings that need to be considered when computing F~μ 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! nn!).