# The Combinatorics of Macdonald's Dn1 Operator

Abstract. To prove the existence of the Macdonald polynomials {P\la(x;q,t)}, \la a partition of n, Macdonald [Séminaire Lotharingien Combin. 20 (1988), Article B20a; "Symmetric functions and Hall polynomials", 2nd ed., Clarendon Press, New York, 1995] introduced an operator Dn1 and proved that for any Schur function s\la(x1, ..., xn), Dn1 s\la(x1, ..., xn) = \sum\mu d\la,\mu(q,t) s\mu(x1, ..., xn) where the sum runs over all partitions \mu of n which are less than or equal to \la in the dominance order and the d\la,\mu(q,t) are polynomials in q and t with integer coefficients. We give an explicit combinatorial formula for the d\la,\mu(q,t)'s.

Received: December 11, 2005. Accepted: September 14, 2006. Final Version: January 5, 2007.

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