A075754 Number of n X n (0,1) matrices containing exactly five 1's in each row and in each column.

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

Offset: 5

Author

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

Keywords

• nonn

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

Crossrefs

Column 5 of A008300.

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)

Extensions

More terms from Brendan McKay, Jan 08 2005