Rectangular arrays

AbstractIn this paper we studied m×n arrays with row sums nr(n,m) and column sums mr(n,m) where (n,m) denotes the greatest common divisor of m and n. We were able to show that the function Hm,n(r), which enumerates m×n arrays with row sums and column sums nr(m,n) and mr(n,m) respectively, is a polynomial in r of degree (m−1)(n−1). We found simple formulas to evaluate these polynomials for negative values, −r, and we show that certain small negative integers are roots of these polynomials. When we considered the generating function Gm,n(y) = Σr−0Hm,n(r)yr, it was found to be rational of degree less than zero. The denominator of Gm,n(y) is of the form (1−y)(m−1)(n−1)+3, and the coefficients of the numerator are non-negative integers which enjoy a certain symmetric relation

Enumeration of arrays of a given size

AbstractEnumeration of arrays whose row and column sums are specified have been studied by a number of people. It has been determined that the function that enumerates square arrays of dimension n, whose rows and columns sum to a fixed non-negative integer r, is a polynomial in r of degree (n − 1)2.In this paper we consider rectangular arrays whose rows sum to a fixed non-negative integer r and whose columns sum to a fixed non-negative integer s, determined by ns = mr. in particular, we show that the functions which enumerate 2 × n and 3 × n arrays with fixed row sums nr(2, n) and nr(3, n), where the symbol (a, b) denotes the greatest common divisor of a and b, and fixed column sums, are polynomials in r of degrees (n − 1) and 2(n − 1) respectively. We have found simple formulas to evaluate these polynomials for negative values, - r, and we show that for certain small negative integers our polynomials will always be zero. We also considered the generating functions of these polynomials and show that they are rational functions of degrees less than zero, whose denominators are of the forms (1 − y)n and (1 − y)2n−1 respectively and whose numerators are polynomials in y whose coefficients satisfy certain properties. In the last section we list the actual polynomials and generating functions in the 2 × n and 3 × n cases for small specific values of n