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

This article is cited in 5 scientific papers (total in 5 papers)

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
Received: 05.05.2001
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


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