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Rectangular arrays with fixed margins. (English) Zbl 0839.05005
Aldous, David (ed.) et al., Discrete probability and algorithms. Proceedings of the workshops “Probability and algorithms” and “The finite Markov chain renaissance” held at IMA, University of Minnesota, Minneapolis, MN, USA, 1993. New York, NY: Springer-Verlag. IMA Vol. Math. Appl. 72, 15-41 (1995).
This is an elegant survey article about the number of rectangular arrays of nonnegative integers with given row and column sums and its importance in various combinatorial and statistical applications. The combinatorial problems include magical squares, enumeration of permutations by descents, enumeration of double cosets, the description of tensor product decompositions, Young tableaux and Kostka numbers. The statistical applications focus on tests for independence in contingency tables with given margins. Several exact and approximative methods for determining the array numbers are described. The computational complexity is discussed. The authors also present Monte Carlo techniques based on Markov chains on the set of arrays with prespecified margins.

MSC:
05A15Exact enumeration problems, generating functions
60G50Sums of independent random variables; random walks
60C05Combinatorial probability
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
05A16Asymptotic enumeration
62H17Contingency tables (statistics)
65C05Monte Carlo methods
05B15Orthogonal arrays, Latin squares, Room squares