n! Theorem

For any partition of , define a polynomial in variables , , ... and , , ... as

(1)

where are the coordinates of the cells of the partition when it is placed in the coordinate plane with base cell at and such that all other coordinates are nonnegative in and . Denote the linear span of all derivatives of this polynomial with respect to the variables by , where represents a partial derivative. This vector space is closed under permutations acting on and simultaneously. Then the theorem states that

(2)

The theorem was proven by M. Haiman in Dec. 1999.

For example, consider the partition . Then

			(3)
			(4)

Then the five derivatives

			(5)
			(6)
			(7)
			(8)
			(9)

together with , elements in all, form a basis for .