On Selecting a Random Shifted Young Tableau

The study of standard and generalized Young tableaux has led to interesting results in quite a number of areas of mathematics. These arrays are of importance in the representation theory of the symmetric and general linear groups [12], in invariant theory [7], and in connection with various combinatorial problems [ 11. In addition, many algorithms have been developed that manipulate the tableaux and their entries [5, 61. We will be particularly concerned with a probabilistic procedure developed by Greene et al. [4]. This algorithm generates a standard Young tableau of fixed shape at random. In so doing, it also provides a proof of the hook formula (Eq. (1.1) below) which enumerates such tableaux. There is another family of arrays, the shifted Young tableaux, that exhibit many similarities to their unshifted cousins [8]. It is the purpose of this paper to show that the Greene, Nijenhuis, and Wilf procedure can be extended to shifted tableaux. Interestingly enough, the algorithms for both types of tableaux are identical, but the proof that all tableaux are equally likely is much more difficult in the shifted case. Let us now make the concepts introduced in the last two paragraphs more precise. A partition of the integer n is a vector X = (x,, X,, . . . , &) having integral components such that A, 2 X, L 2 & > 0 and Zi& = n. The shape of X, S, is an array of n cells or nodes into r left-justified rows with 4 cells in row i. Finally by placing the numbers 1,2, . . . , n into the cells of S so that the rows and columns increase we obtain a standard Young Tableau of shape S (see Fig. 1). The number of standard Young tableaux of shape S will be denoted by&