[Paper Review] Asymptotic enumeration of contingency tables with constant margins

E. Rodney Canfield, Brendan D. McKay|arXiv (Cornell University)|Mar 20, 2007

Advanced Combinatorial Mathematics21 references10 citations

TL;DR

This paper establishes a precise asymptotic formula for the number of m × n contingency tables with fixed row and column sums, valid when m, n grow large and remain approximately equal, with bounded ratios and non-vanishing average entries. The result generalizes magic squares and semiregular bipartite multigraphs, providing a unified asymptotic estimate across a broad parameter range.

ABSTRACT

Let s,t,m,n be positive integers such that sm=tn. Let M(m,s;n,t) be the number of m x 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 x 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 x n magic 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 become infinite 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.

Motivation & Objective

• To derive an asymptotic formula for the number of m × n matrices with non-negative integer entries, fixed row sums s, and fixed column sums t.
• To extend existing results on magic squares and bipartite multigraphs to a broader class of contingency tables with balanced marginal constraints.
• To determine the range of parameters (m, s; n, t) for which the asymptotic formula remains valid, particularly when m, n → ∞ and m/n, n/m, s/n, t/m are bounded below.

Proposed method

• Utilizes combinatorial and probabilistic techniques, including the configuration model and multivariate saddle-point methods, to analyze the structure of contingency tables.
• Applies generating functions and Laplace's method to approximate the number of integer solutions to the marginal sum constraints.
• Employs symmetry and scaling arguments to reduce the problem to a limiting form under the condition sm = tn.
• Derives an asymptotic expression based on the volume of the transportation polytope and the density of integer points within it.
• Considers the average entry size as a key parameter, ensuring it remains bounded away from zero to maintain asymptotic validity.
• Validates the formula's range by analyzing the behavior of the logarithmic number of tables under scaling limits of m, n, s, t.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of m × n contingency tables with fixed row and column sums when m and n grow large?
• RQ2 How does the asymptotic formula behave when the row and column sums are proportional and the average entry remains bounded away from zero?
• RQ3 Over what range of parameters (m, s; n, t) is the asymptotic formula valid, particularly when m/n and s/t are bounded?
• RQ4 Can the formula be extended to include magic squares and semiregular bipartite multigraphs as special cases?
• RQ5 What is the role of the average entry size in determining the accuracy of the asymptotic approximation?

Key findings

• The paper derives a precise asymptotic formula for M(m,s;n,t) that holds when m, n → ∞ and the ratios m/n, n/m, s/n, t/m are bounded from below.
• The formula applies to all cases where sm = tn, including magic squares (when m = n and s = t) and semiregular bipartite multigraphs.
• The asymptotic estimate is valid even when the average entry is not large, provided it remains bounded away from zero.
• The result generalizes and refines prior work on magic squares and contingency tables by extending the range of applicability.
• The formula captures the exponential growth rate of contingency tables through a combination of volume and entropy terms in the logarithmic scale.
• The method ensures that the approximation error is controlled uniformly across the specified parameter regime.