A combinatorial model for the Macdonald polynomials

We introduce a polynomial (C) over tilde (mu)[Z; q, t], depending on a set of variables Z = z(1), z(2), . . . , a partition mu, and two extra parameters q, t. The definition of (C) over tilde (mu) involves a pair of statistics (maj(sigma, mu), inv(sigma, mu)) on words sigma of positive integers, and the coefficients of the z(i) are manifestly in N[q, t]. We conjecture that (C) over tilde (mu)[Z; q, t] is none other than the modified Macdonald polynomial (H) over tilde (mu)[Z; q, t]. We further introduce a general family of polynomials F-T[Z; q, S], where T is an arbitrary set of squares in the first quadrant of the xy plane, and S is an arbitrary subset of T. The coefficients of the F-T[Z; q, S] are in N[q], and (C) over tilde (mu)[Z; q, t] is a sum of certain F-T[Z; q, S] times nonnegative powers of t. We prove F-T[Z; q, S] is symmetric in the z(i) and satisfies other properties consistent with the conjecture. We also show how the coefficient of a monomial in F-T[Z; q, S] can be expressed recursively. MAPLE calculations indicate the F-T[Z; q, S] are Schur-positive, and we present a combinatorial conjecture for their Schur coefficients when the set T is a partition with at most three columns.