Rectangular Arrays with Fixed Margins

Abstract

In a variety of combinatorial and statistical applications, one needs to know the number of rectangular arrays of nonnegative integers with given row and column sums. The combinatorial problems include counting magic squares, enumerating permutations by descent patterns and a variety of problems in representation theory. The statistical problems involve goodness of fit tests for contingency tables. We review these problems along with the available techniques for exact and approximate solution.