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 Diskr. Mat., 2002, Volume 14, Issue 4, Pages 65–86 (Mi dm264)

On permanents of random doubly stochastic matrices and on asymptotic estimates for the number of Latin rectangles and Latin squares

A. N. Timashev

Abstract: We consider the class $\mathfrak A_n(k)$ of all $(0,1)$-matrices $A_k$ of size $n\times n$ with exactly $k$ ones in each row and each column, $k=1,…,n$. We prove an asymptotic formula for the permanent $\operatorname{per}A_k$, which holds true as $n\to\infty$ and $0<n-k=o(n/\ln n)$ uniformly with respect to $A_k\in\mathfrak A_n(k)$. We discuss the known upper and lower bounds for the numbers of $m\times n$ Latin rectangles and of $n\times n$ Latin squares and asymptotic expressions of these numbers as $n\to\infty$ and $m=m(n)$. We notice that the well-known O'Neil conjecture on the asymptotic behaviour of the number of Latin squares holds in a strong form. We formulate new conjectures of such kind and deduce from these conjectures asymptotic estimates of the numbers of Latin rectangles and Latin squares that sharpen the results known before. In conclusion, we give a short review of the literature devoted to the questions discussed in the paper with formulations of the main results.

DOI: https://doi.org/10.4213/dm264

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English version:
Discrete Mathematics and Applications, 2002, 12:5, 431–452

Bibliographic databases:

UDC: 519.2
Revised: 14.02.2002

Citation: A. N. Timashev, “On permanents of random doubly stochastic matrices and on asymptotic estimates for the number of Latin rectangles and Latin squares”, Diskr. Mat., 14:4 (2002), 65–86; Discrete Math. Appl., 12:5 (2002), 431–452

Citation in format AMSBIB
\Bibitem{Tim02} \by A.~N.~Timashev \paper On permanents of random doubly stochastic matrices and on asymptotic estimates for the number of Latin rectangles and Latin squares \jour Diskr. Mat. \yr 2002 \vol 14 \issue 4 \pages 65--86 \mathnet{http://mi.mathnet.ru/dm264} \crossref{https://doi.org/10.4213/dm264} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1964121} \zmath{https://zbmath.org/?q=an:1048.05016} \transl \jour Discrete Math. Appl. \yr 2002 \vol 12 \issue 5 \pages 431--452 

• http://mi.mathnet.ru/eng/dm264
• https://doi.org/10.4213/dm264
• http://mi.mathnet.ru/eng/dm/v14/i4/p65

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1. F. I. Solov'eva, A. V. Los', “On partitions into perfect $q$-ary codes”, J. Appl. Industr. Math., 4:1 (2010), 136–142
2. Greenhill C., McKay B.D., “Random Dense Bipartite Graphs and Directed Graphs With Specified Degrees”, Random Structures & Algorithms, 35:2 (2009), 222–249
3. Cameron P.J., “A generalisation of t-designs”, Discrete Mathematics, 309:14 (2009), 4835–4842
4. Stones D.S., “The many formulae for the number of Latin rectangles”, Electronic Journal of Combinatorics, 17:1 (2010), A1
5. Kocharovsky V.V., Kocharovsky V.V., “On the permanents of circulant and degenerate Schur matrices”, Linear Alg. Appl., 519 (2017), 366–381
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