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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 (list; graph; refs; listen; history; text; internal format)
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*s-1/2*(s-1)^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) 207-221.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 5..61, (computed with program by Doron Zeilberger, see link below)

B. D. McKay, 0-1 matrices with constant row and column sums

E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 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 LA-4434, 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^(1/2-6*n)*5^(3*n+1/2) *9^(-n)*exp(-5*n-8)*n^(5*n+1/2)

(End)

CROSSREFS

Column 5 of A008300.

Sequence in context: A227668 A010799 A283830 * A143476 A008979 A158044

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

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Last modified May 23 12:45 EDT 2017. Contains 286915 sequences.