Improved estimates for the number of non-negative integer matrices with given row and column sums-Reference-Cited by-同舟云学术

Improved estimates for the number of non-negative integer matrices with given row and column sums

Published:2024-01 Issue:2282 Volume:480 Page:
ISSN:1364-5021
Container-title:Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
language:en
Short-container-title:Proc. R. Soc. A.

Author:

Jerdee Maximilian¹^ORCID,Kirkley Alec²^ORCID,Newman M. E. J.¹³^ORCID

Affiliation:

1. Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

2. Institute of Data Science, University of Hong Kong, Hong Kong SAR, China

3. Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA

Abstract

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.

Funder

National Science Foundation

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

Link

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2023.0470

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