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A058528 Number of n X n (0,1) matrices with all column and row sums equal to 4. 6
1, 0, 0, 0, 1, 120, 67950, 68938800, 116963796250, 315031400802720, 1289144584143523800, 7722015017013984456000, 65599839591251908982712750, 769237071909157579108571190000, 12163525741347497524178307740904300 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
Further terms generated by a Mathematica program written by Gordon G. Cash, who thanks B. R. Perez-Salvador, Universidad Autonoma Metropolitana Unidad Iztapalapa, Mexico, for providing the algorithm on which this program was based.
Also number of ways to arrange 4n rooks on an n X n chessboard, with no more than 4 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. R. Perez-Salvador, S. de los Cobos Silva, M. A. Gutierrez-Andrade and A. Torres-Chazaro, A Reduced Formula for Precise Numbers of (0,1) Matrices in a(R,S), Disc. Math., 2002, 256, 361-372.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..150, [Computed with Maple program by Doron Zeilberger, see link below. This replaces an earlier b-file computed by Vladeta Jovovic (and corrected terms 26-31).]
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; Local copy [Pdf file only, no active links]
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]
FORMULA
a(n) = 24^{-n} sum_{alpha +beta + gamma + mu + u =n}frac{3^{ gamma }(-6)^{beta +u }8^{ mu }(n!)^{2}(4alpha +2 gamma + mu )!(beta +2 gamma )!}{alpha!beta! gamma! mu!u!} sum_{i=0}^{ floor (beta +2 gamma )/2 }frac{1}{24^{alpha - gamma +i}2^{beta +2 gamma -i}i!(beta +2 gamma -2i)!(alpha - gamma +i)!} - Shanzhen Gao, Nov 07 2007
From Vaclav Kotesovec, Aug 04 2013: (Start)
a(n) ~ exp(-1/2)*C(n,4)^(2*n)/C(n^2,4*n), (Canfield + McKay, 2004).
a(n) ~ sqrt(Pi)*2^(2*n+3/2)*9^(-n)*exp(-4*n-9/2)*n^(4*n+1/2).
(End)
EXAMPLE
a(4) = 1 because there is only one possible 4 X 4 (0,1) matrix with all row and column sums equal to 4, the matrix of all 1's. a(5) = 120 = 5! because there are 5X4X3X2X1 ways of placing a zero in each successive column (row) so that it is not in the same row (column) as any previously placed.
CROSSREFS
Column 4 of A008300. Row sums of A284991.
Sequence in context: A065961 A364512 A333043 * A001421 A107446 A184887
KEYWORD
nonn
AUTHOR
David desJardins, Dec 22 2000
EXTENSIONS
More terms from Gordon G. Cash (cash.gordon(AT)epa.gov), Oct 22 2002
More terms from Vladeta Jovovic, Nov 12 2006
STATUS
approved

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