A104602 - OEIS

%I #27 Nov 15 2018 08:39:50

%S 1,1,2,10,70,642,7246,97052,1503700,26448872,520556146,11333475922,

%T 270422904986,7016943483450,196717253145470,5925211960335162,

%U 190825629733950454,6543503207678564364,238019066600097607402,9153956822981328930170,371126108428565106918404

%N Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns.

%C Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns, up to row and column permutation, is A057151(n). - _Vladeta Jovovic_, Mar 25 2006

%H Alois P. Heinz, <a href="/A104602/b104602.txt">Table of n, a(n) for n = 0..400</a>

%H H. Cheballah, S. Giraudo, R. Maurice, <a href="http://arxiv.org/abs/1306.6605">Combinatorial Hopf algebra structure on packed square matrices</a>, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.

%H M. Maia and M. Mendez, <a href="https://arxiv.org/abs/math/0503436">On the arithmetic product of combinatorial species</a>, arXiv:math/0503436 [math.CO], 2005.

%F a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A048144(k). - _Vladeta Jovovic_, Mar 25 2006

%F G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^n. - _Vladeta Jovovic_, Mar 25 2006

%F a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.28889864564457451375789435201798... . - _Vaclav Kotesovec_, May 07 2014

%F In closed form, c = 1 / (log(2) * 2^(log(2)/2+2) * sqrt(Pi*(1-log(2)))). - _Vaclav Kotesovec_, May 03 2015

%F G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n*(n+1)). - _Paul D. Hanna_, Mar 26 2018

%e From _Gus Wiseman_, Nov 14 2018: (Start)

%e The a(3) = 10 matrices:

%e   [1 1] [1 1] [1 0] [0 1]

%e   [1 0] [0 1] [1 1] [1 1]

%e .

%e   [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]

%e   [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]

%e   [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

%e (End)

%t Table[1/n!*Sum[StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}],{k,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, May 07 2014 *)

%t Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* _Gus Wiseman_, Nov 14 2018 *)

%Y Row sums of triangle A104601.

%Y Cf. A048291, A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.

%K nonn

%O 0,3

%A _Ralf Stephan_, Mar 27 2005

%E More terms from _Vladeta Jovovic_, Mar 25 2006

%E a(0)=1 prepended by _Alois P. Heinz_, Jan 14 2015