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A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.
(Formerly M4286 N1792)
22
1, 0, 1, 6, 90, 2040, 67950, 3110940, 187530840, 14398171200, 1371785398200, 158815387962000, 21959547410077200, 3574340599104475200, 676508133623135814000, 147320988741542099484000, 36574751938491748341360000, 10268902998771351157327104000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Or, number of labeled 2-regular relations of order n.

Also number of ways to arrange 2n rooks on an n X n chessboard, with no more than 2 rooks in each row and column (no 3 in a line). - Vaclav Kotesovec, Aug 03 2013

REFERENCES

R. Bricard, L'Intermédiaire des Mathématiciens, 8 (1901), 312-313.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,2).

L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review. Solution by D. E. Knuth. Reprinted in Problems in Applied Mathematics, ed. M. Klamkin, SIAM, 1990, p. 350.

Gao, Shanzhen, and Matheis, Kenneth, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.

J. T. Lewis, Maximal L-free subsets of a squarefree array, Congressus Numerantium, 141 (1999), 151-155.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (b).

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.

J. H. van Lint and R. M. Wilson, A Course in Combinatorics (Cambridge University Press, Cambridge, 1992), pp. 152-153. [The second edition is said to be a better reference.]

LINKS

R. W. Robinson and Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..48 from R. W. Robinson)

H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769.

P. Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806.

L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782.

Sally Cockburn and Joshua Lesperance, Deranged Socks, Mathematics Magazine, Vol. 86, no. 2, April 2013, pp. 97-109.

M. E. Kuczma, 0-1-Matrices with Line-Sums Equal to 2, Am. Math. Month. 99 (1992) 959-961, E3419.

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]

Bo-Ying Wang, Fuzhen Zhang, On the precise number of (0,1)-matrices in A(R,S), Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From N. J. A. Sloane, Jun 07 2012

Index entries for sequences related to binary matrices

FORMULA

a(n) = (n! (n-1) Gamma(n-1/2) / Gamma(1/2) ) * 1F1[2-n; 3/2-n; -1/2] [Erlebach and Ruehr]. This representation is exact, asymptotic and convergent.

2*a(n) -2*n*(n-1)*a(n-1) -n*(n-1)^2*a(n-2)=0.

a(n) ~ 2 sqrt(Pi) n^(2n + 1/2) e^(-2n - 1/2) [Knuth]

a(n) = (1/2)*n*(n-1)^2 * ( (2*n-3)*a(n-2) + (n-2)^2*a(n-3) ) (from Anand et al.)

Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(-x/2)/sqrt(1-x); a(n) = n(n-1)/2 [ 2 a(n-1) + (n-1) a(n-2) ] (Bricard)

b_n = a_n/n! satisfies b_n = (n-1)(b_{n-1} + b_{n-2}/2); e.g.f. for {b_n} and for derangements (A000166) are related by D(x) = B(x)^2.

lim(n->infinity) sqrt(n)*a(n)/(n!)^2 = A096411 [Kuczma]. - R. J. Mathar, Sep 21 2007

a(n) = 4^(-n) * n!^2 * sum(i=0..n, (-2)^i * (2*n-2*i)! / (i!*(n-i)!^2)) ). - Shanzhen Gao, Feb 15 2010

MATHEMATICA

a[n_] := (n-1)*n!*Gamma[n-1/2]*Hypergeometric1F1[2-n, 3/2-n, -1/2]/Sqrt[Pi]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 06 2011, after first formula *)

PROG

(PARI) a(n)=if(n<2, n==0, (n^2-n)*(a(n-1)+(n-1)/2*a(n-2)))

(Haskell)

a001499 n = a001499_list !! n

a001499_list = 1 : 0 : 1 : zipWith (*) (drop 2 a002411_list)

   (zipWith (+) (zipWith (*) [3, 5 ..] $ tail a001499_list)

                (zipWith (*) (tail a000290_list) a001499_list))

-- Reinhard Zumkeller, Jun 02 2013

CROSSREFS

Cf. A000681, A053871, A123544 (connected relations), A000986 (symmetric matrices).

Cf. A000290, A002411, A005408.

Cf. A001501. Column 2 of A008300. Row sums of A284989.

Sequence in context: A002896 A266734 A004996 * A147630 A221097 A177584

Adjacent sequences:  A001496 A001497 A001498 * A001500 A001501 A001502

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 27 11:25 EDT 2017. Contains 288788 sequences.