A new recursion for three-column combinatorial Macdonald polynomials

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 ( 3 n ) ! to ( 3 n ) ! / ( 3 ! n n ! ) .