On permanents of random doubly stochastic matrices and asymptotic estimates of the nom hers of Latin rectangles and Latin squares
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A.N. Timashov
Abstract
We consider the class An(k) of all (0, 1)-matrices Ak of size n × n with exactly k ones in each row and each column, k = 1, ... , n. We prove an asymptotic formula for the permanent per Ak, which holds true as n → ∞ and 0 < n - k = o(n/ln n) uniformly with respect to Ak ∈ An(k). We discuss the known upper and lower bounds for the numbers of m × n Latin rectangles and of n × n Latin squares and asymptotic expressions of these numbers as n → ∞ 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.
© 2016 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Editorial
- On permanents of random doubly stochastic matrices and asymptotic estimates of the nom hers of Latin rectangles and Latin squares
- On reversal of graph orientation
- The structure of an infinite Sylow subgroup in some periodic Shunkov groups
- Boolean lattices of multiply Ω-foliated formations
- Boolean lattices of n-multiply Ω-bicanonical Fitting classes
- On infinite groups with points
- Linear recurring sequence decimations with exponentially increasing step
- On ω-languages of special billiards
- On relative complexity of quantum and classical branching programs
- Forthcoming Papers
- Contents
Articles in the same Issue
- Editorial
- On permanents of random doubly stochastic matrices and asymptotic estimates of the nom hers of Latin rectangles and Latin squares
- On reversal of graph orientation
- The structure of an infinite Sylow subgroup in some periodic Shunkov groups
- Boolean lattices of multiply Ω-foliated formations
- Boolean lattices of n-multiply Ω-bicanonical Fitting classes
- On infinite groups with points
- Linear recurring sequence decimations with exponentially increasing step
- On ω-languages of special billiards
- On relative complexity of quantum and classical branching programs
- Forthcoming Papers
- Contents
