A New Derivation of the Counting Formula for Young Tableaux

Given integers a i > . . . > a, 2 0 the corresponding Young tableau is that part of an a, x n matrix consisting of all (i,j) entries with n >j > 1, aj > i > 1. The problem which we discuss here is: In how many ways can the integers l,..., Cy ai be placed into the entries of the a, ,..., a, Young tableau so that any row and any column forms an increasing sequence? We call this number F(ai ,..., a,) and we give a new formula for F(u, ,..., a,). The known solution for this problem (see [ 1, 21 for more details) is by the so-called hook formula which can be easily derived from our formula. Notice that F(u, ,..., a,) is the unique solution for difference equation (1) under initial conditions (2~(4).