ROOK EQUIVALENCE OF FERRERS BOARDS

We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board. Introduction. Let N denote the set of positive integers. A board is a finite subset of N x N. Intuitively, a board is an array of squares or cells arranged in rows and columns, i.e., a board is a subset of the set of squares of an n x n chessboard. We shall frequently utilize this intuitive terminology. We consider two boards to be the same board if one is a translate of the other, i.e., boards B and B' are the same if there exist integers a and b such that B' = (i + a, i + b): (i, j) E B}. Thus, only the geometric configuration or the relative positions of the cells is of significance. For a board B, IBI denotes the number of cells in B. Let rk be the number of ways of placing k nontaking rooks (no two in the same row or column) on the board B, i.e., the number of k-subsets of the set B such that no two elements of a k-subset have the same first component or the same second component. When no confusion can arise we suppress the B and write rk. The rook vector of a board B is defined to be the vector r(B) (ro, r1, r2, .. . ) where r = 1. Note that from some point on all the re's are zero, in particular r. = 0 for i > BI. Two boards are called rook equivalent if they have the same rook vector. A board B is a Ferrers board if there exists a nondecreasing finite sequence of positive integers hi, h2. hc such that B = 1(i, j): i < c and j < hi}. Intuitively, a Ferrers board is a board made up of adjacent solid columns of cells with a common lower edge and such that the height of the columns from left to right forms a nondecreasing sequence. If the heights of the columns form a strictly increasing sequence, then we call the board an increasing Ferrers board. Examples of a Ferrers board and an increasing Ferrers board are given in Figures l(a) and l(b), respectively. Received by the editors August 21, 1974. AMS (MOS) subject classifications (1970). Primary 05A10, 05A15, 05A19.