

A075754


Number of n X n (0,1) matrices containing exactly five 1's in each row and in each column.


4



1, 720, 3110940, 24046189440, 315031400802720, 6736218287430460752, 226885231700215713535680, 11649337108041078980732943360, 885282776210120715086715619724160, 96986285294151066094112970262797953280
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OFFSET

5,2


COMMENTS

Also number of ways to arrange 5n rooks on an n X n chessboard, with no more than 5 rooks in each row and column.  Vaclav Kotesovec, Aug 04 2013
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*s1/2*(s1)^2)*(n*s)^(n*s+1/2)*(s!)^(2*n).  Vaclav Kotesovec, Aug 04 2013


REFERENCES

B. D. McKay, Applications of a technique for labeled enumeration, Congressus Numerantium, 40 (1983) 207221.


LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 5..61, (computed with program by Doron Zeilberger, see link below)
B. D. McKay, 01 matrices with constant row and column sums
E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 01 matrices with equal row and column sums.
Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses, Maple package Bipartite.
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Index entries for sequences related to binary matrices


FORMULA

From Vaclav Kotesovec, Aug 04 2013: (Start)
a(n) ~ exp(1/2)*binomial(n,5)^(2*n) / binomial(n^2,5*n), (Canfield + McKay, 2004)
a(n) ~ sqrt(Pi)*2^(1/26*n)*5^(3*n+1/2) *9^(n)*exp(5*n8)*n^(5*n+1/2)
(End)


CROSSREFS

Column 5 of A008300.
Sequence in context: A227668 A010799 A283830 * A318711 A143476 A008979
Adjacent sequences: A075751 A075752 A075753 * A075755 A075756 A075757


KEYWORD

nonn


AUTHOR

Michel Buffet (buffet(AT)engref.fr), Oct 08 2002


EXTENSIONS

More terms from Brendan McKay, Jan 08 2005


STATUS

approved



