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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).] 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; Local copy [Pdf file only, no active links] B. D. McKay, 0-1 matrices with constant row and column sums 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 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 Adjacent sequences: A058525 A058526 A058527 * A058529 A058530 A058531 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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