An inversion formula involving partitions

In this note we outline a combinatorial proof of an inversion formula involving partitions of a number. This formula can be used to obtain the theory of symmetric group characters in a purely combinatorial way, as will be done in a forthcoming book, The combinatorics of the symmetric group, by the present author and Dr. G.-C. Rota. The terminology we use is as follows. By a composition a of an integer n we mean a sequence (a1? a2,..., as) of nonnegative integers whose sum is n. A partition of n is a composition X = (Xu A2,..., Xp) with Xx^ X2 ^ • • • ̂ Xp > 0. The notation Xv—n means "A is a partition of n". We use the symbols a, /? for compositions, X, \i, p for partitions. A Young diagram of shape X is an array of dots, with Xt dots in the first row, X2 in the second row, etc., in which the first dots from the rows lie in a column, the second dots form a column, and so on. The conjugate partition X of X is the shape obtained when the Young diagram of shape X is transposed about its main diagonal, i.e., the rows of the transposed diagram are the columns of the original diagram. A generalized Young tableau (GYT) n of shape X is an array of integers qtj(i = 1,2, . . . ,p , j = 1,2,...,Af) with qtJ > 09qiJ+1 ^ qtJ if j < A„ and qi+1J > qtJ if j ^ Xi + 1, i.e., an array of positive integers of shape X which is increasing nonstrictly along the rows and increasing strictly down the columns. The type of a GYT n is the composition a = (als a2,..., as) of n (where X t— n\ where af is the number of times the integer i appears in n. If a = (a ! , . . . , as) is a composition of n with s ^ n , and TES„ (the symmetric group on {1,2,..., n}\ then T • a is the composition of n whose parts are af + T(0 — i, i = 1,2,..., n (where af = 0 if i > s), if all these parts are nonnegative, and T • a is undefined otherwise. We also define T * X to be the partition of n whose parts are Xt + x(i) — i in nonincreasing order if all these parts are nonnegative, and T*A is undefined otherwise. Our inversion formula can now be stated.