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Asymptotic enumeration of integer matrices with large equal row and column sums

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Abstract

Let s, t, m, n be positive integers such that sm = tn. Let M(m, s; n, t) be the number of m×n matrices over {0, 1, 2, …} with each row summing to s and each column summing to t. Equivalently, M(m, s; n, t) counts 2-way contingency tables of order m×n such that the row marginal sums are all s and the column marginal sums are all t. A third equivalent description is that M(m, s; n, t) is the number of semiregular labelled bipartite multigraphs with m vertices of degree s and n vertices of degree t. When m = n and s = t such matrices are also referred to as n×n magic or semimagic squares with line sums equal to t. We prove a precise asymptotic formula for M(m, s; n, t) which is valid over a range of (m, s; n, t) in which m, n→∞ while remaining approximately equal and the average entry is not too small. This range includes the case where m/n, n/m, s/n and t/m are bounded from below.

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Authors and Affiliations

  1. Department of Computer Science, University of Georgia, Athens, GA, 30602, USA

    E. Rodney Canfield

  2. Department of Computer Science, Australian National University, Canberra, ACT, 0200, Australia

    Brendan D. McKay

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  1. E. Rodney Canfield
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  2. Brendan D. McKay
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Correspondence to E. Rodney Canfield.

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Research supported by the NSA Mathematical Sciences Program.

Research supported by the Australian Research Council.

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Rodney Canfield, E., McKay, B.D. Asymptotic enumeration of integer matrices with large equal row and column sums. Combinatorica 30, 655–680 (2010). https://doi.org/10.1007/s00493-010-2426-1

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  • DOI: https://doi.org/10.1007/s00493-010-2426-1

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