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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Counting binary matrices with given row and column sums
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by Ben Johnsen and Eldar Straume PDF
Math. Comp. 48 (1987), 737-750 Request permission


This paper is concerned with the calculation of certain numbers $sb(\bar p)$, $b(\bar p,\bar q)$ related to combinatorial problems and graph theory. $\bar p,\bar q$ are vectors of nonnegative integers, and $sb(\bar p)$ is the number of labelled graphs with vertex degree sequence $\bar p$, or equivalently, the number of 0-diagonal, symmetric, binary matrices with row sum $\bar p$. Similarly, $b(\bar p,\bar q)$ is the number of binary rectangular matrices with row sum $\bar p$ and column sum $\bar q$. The numbers also appear as coefficients in the expansions $\Pi (1 + {x_i}{x_j})$, $\Pi (1 + {x_i}{y_j})$. Explicit (i.e., nonrecursive) formulas for $sb(\bar p)$, $b(\bar p,\bar q)$ are developed, together with an analysis of their complexity. Properties of $\bar p$ (or $\bar q$), such as $\max \{ {p_i}\}$, $k = \Sigma {p_i}$, $n = \# \{ {p_i} \ne 0\}$, are incorporated into a numerical invariant which measures the total "cost" (or computing time). The performance of the theory for practical calculations has been thoroughly tested. For example, with suitable restrictions on $\bar p$ one may obtain $sb(\bar p)$ for, say, $n = 20$ or $k = 30$.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 48 (1987), 737-750
  • MSC: Primary 05A15; Secondary 05B20, 05C30
  • DOI:
  • MathSciNet review: 878703