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A104602 Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns. 20
1, 1, 2, 10, 70, 642, 7246, 97052, 1503700, 26448872, 520556146, 11333475922, 270422904986, 7016943483450, 196717253145470, 5925211960335162, 190825629733950454, 6543503207678564364, 238019066600097607402, 9153956822981328930170, 371126108428565106918404 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400

H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.

M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

FORMULA

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A048144(k). - Vladeta Jovovic, Mar 25 2006

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

a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.28889864564457451375789435201798... . - Vaclav Kotesovec, May 07 2014

In closed form, c = 1 / (log(2) * 2^(log(2)/2+2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015

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

EXAMPLE

From Gus Wiseman, Nov 14 2018: (Start)

The a(3) = 10 matrices:

  [1 1] [1 1] [1 0] [0 1]

  [1 0] [0 1] [1 1] [1 1]

.

  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]

  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]

  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

(End)

MATHEMATICA

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 *)

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

CROSSREFS

Row sums of triangle A104601.

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

Sequence in context: A289680 A089845 A293962 * A320957 A341477 A317980

Adjacent sequences:  A104599 A104600 A104601 * A104603 A104604 A104605

KEYWORD

nonn

AUTHOR

Ralf Stephan, Mar 27 2005

EXTENSIONS

More terms from Vladeta Jovovic, Mar 25 2006

a(0)=1 prepended by Alois P. Heinz, Jan 14 2015

STATUS

approved

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Last modified August 4 09:47 EDT 2021. Contains 346446 sequences. (Running on oeis4.)