This site is supported by donations to The OEIS Foundation.



The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.
(Formerly M4286 N1792)

%I M4286 N1792

%S 1,0,1,6,90,2040,67950,3110940,187530840,14398171200,1371785398200,

%T 158815387962000,21959547410077200,3574340599104475200,

%U 676508133623135814000,147320988741542099484000,36574751938491748341360000,10268902998771351157327104000

%N Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.

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

%C 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

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

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

%D 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.

%D 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.

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

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

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

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

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

%D 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.

%D 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.]

%H R. W. Robinson and Alois P. Heinz, <a href="/A001499/b001499.txt">Table of n, a(n) for n = 0..200</a> (terms n = 0..48 from R. W. Robinson)

%H H. Anand, V. C. Dumir and H. Gupta, <a href="http://dx.doi.org/10.1215/S0012-7094-66-03391-6">A combinatorial distribution problem</a>, Duke Math. J., 33 (1996), 757-769.

%H P. Barry, <a href="http://dx.doi.org/10.1155/2013/657806">On the Connection Coefficients of the Chebyshev-Boubaker polynomials</a>, The Scientific World Journal, Volume 2013 (2013), Article ID 657806.

%H L. Carlitz, <a href="http://dx.doi.org/10.1215/S0012-7094-66-03392-8">Enumeration of symmetric arrays</a>, Duke Math. J., Vol. 33 (1966), 771-782.

%H Sally Cockburn and Joshua Lesperance, <a href="http://www.maa.org/sites/default/files/pdf/upload_library/2/Cockburn-MathMag-2014.pdf">Deranged Socks</a>, Mathematics Magazine, Vol. 86, no. 2, April 2013, pp. 97-109.

%H M. E. Kuczma, <a href="http://www.jstor.org/stable/2324501">0-1-Matrices with Line-Sums Equal to 2</a>, Am. Math. Month. 99 (1992) 959-961, E3419.

%H M. L. Stein and P. R. Stein, <a href="/A001496/a001496.pdf">Enumeration of Stochastic Matrices with Integer Elements</a>, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

%H Bo-Ying Wang, Fuzhen Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00197-0">On the precise number of (0,1)-matrices in A(R,S)</a>, Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From _N. J. A. Sloane_, Jun 07 2012

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%F 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.

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

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

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

%F 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)

%F 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.

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

%F 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

%t 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 *)

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

%o (PARI) seq(n)={Vec(serlaplace(serlaplace(exp(-x/2 + O(x*x^n))/sqrt(1-x + O(x*x^n)))))}; \\ _Andrew Howroyd_, Sep 09 2018

%o (Haskell)

%o a001499 n = a001499_list !! n

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

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

%o (zipWith (*) (tail a000290_list) a001499_list))

%o -- _Reinhard Zumkeller_, Jun 02 2013

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

%Y Cf. A000290, A002411, A005408.

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

%K nonn,nice,easy

%O 0,4

%A _N. J. A. Sloane_

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 25 01:17 EDT 2018. Contains 315360 sequences. (Running on oeis4.)