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On the number of equivalence classes of \((0,1)\)-matrices. (English. Russian original) Zbl 0827.15014
Russ. Acad. Sci., Dokl., Math. 48, No. 3, 461-464 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 333, No. 2, 144-146 (1993).
Let us consider \((0,1)\)-matrices of size \(m \times n\). Two matrices are regarded to be equivalent, if one of them can be obtained from the other by permuting rows and columns. Clearly, all matrices of one class have the same number of ones. Let \(S^\nu_{m,n}\) be the number of equivalence classes of \((0,1)\)-matrices of size \(m \times n\) with \(\nu\) ones.
The author gives various estimates on \(S^\nu_{m,n}\). For example, if \(S_{m,n} = \sum^{mn}_{\nu = 0} S^\nu_{m,n}\), then \(S_{m,n} = {2^{mn} \over m! n!} (1 + O (R_{m,n}))\), where \(R_{m,n} = {m^2 \over 2^{n/2}} + {n^2 \over 2^{m/2}} \to 0\) if \(n,m \to \infty\).
15B36 Matrices of integers
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)