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A172544 Number of n X n 0..1 arrays with row sums 6 and column sums 6.

%I A172544 #17 Jun 02 2025 02:24:12
%S A172544 0,0,0,0,0,1,5040,187530840,12025780892160,1289144584143523800,
%T A172544 226885231700215713535680,64051375889927380035549804336,
%U A172544 28278447454165011203551734584421120
%N A172544 Number of n X n 0..1 arrays with row sums 6 and column sums 6.
%C A172544 Also number of ways to arrange 6n rooks on an n X n chessboard, with no more than 6 rooks in each row and column. - _Vaclav Kotesovec_, Aug 04 2013
%C A172544 Generally (Canfield + McKay, 2004), a(n) ~ exp(-1/2)*binomial(n,s)^(2*n) / binomial(n^2,s*n), or a(n) ~ sqrt(2*Pi)*exp(-n*s-1/2*(s-1)^2)*(n*s)^(n*s+1/2)*(s!)^(-2*n). - _Vaclav Kotesovec_, Aug 04 2013
%H A172544 R. H. Hardin, <a href="/A172544/b172544.txt">Table of n, a(n) for n=1..28</a>
%H A172544 E. R. Canfield and B. D. McKay, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r29">Asymptotic enumeration of dense 0-1 matrices with equal row and column sums</a>, Electron. J. Combin. 12 (2005)
%H A172544 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/bipartite.html">In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses</a>
%F A172544 From _Vaclav Kotesovec_, Aug 04 2013: (Start)
%F A172544 a(n) ~ exp(-1/2)*binomial(n,6)^(2*n)/binomial(n^2,6*n), (Canfield + McKay, 2004)
%F A172544 a(n) ~ sqrt(Pi)*2^(1-2*n)*3^(2*n+1/2)*5^(-2*n)*exp(-6*n-25/2)*n^(6*n+1/2)
%F A172544 (End)
%Y A172544 Column 6 of A008300.
%K A172544 nonn
%O A172544 1,7
%A A172544 _R. H. Hardin_ Feb 06 2010