A101370 Number of zero-one matrices with n ones and no zero rows or columns.
1, 4, 24, 196, 2016, 24976, 361792, 5997872, 111969552, 2324081728, 53089540992, 1323476327488, 35752797376128, 1040367629940352, 32441861122796672, 1079239231677587264, 38151510015777089280, 1428149538870997774080, 56435732691153773665280
Offset: 1
Examples
a(2)=4: [1 1] [1] [1 0] [0 1] ..... [1] [0 1] [1 0] From _Gus Wiseman_, Nov 14 2018: (Start) The a(3) = 24 matrices: [111] . [11][11][110][101][10][100][011][01][010][001] [10][01][001][010][11][011][100][11][101][110] . [1][10][10][10][100][100][01][01][010][01][010][001][001] [1][10][01][01][010][001][10][10][100][01][001][100][010] [1][01][10][01][001][010][10][01][001][10][100][010][100] (End)
References
- Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, p. 435 (IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13), Springer, Berlin. [Rainer Rosenthal, Apr 10 2007]
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..400
- P. J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
- Giulio Cerbai and Anders Claesson, Enumerative aspects of Caylerian polynomials, arXiv:2411.08426 [math.CO], 2024. See pp. 3, 19.
- Loïc Foissy, Claudia Malvenuto, and Frédéric Patras, Matrix symmetric and quasi-symmetric functions and noncommutative representation theory, arXiv:2503.14417 [math.CO], 2025. See p. 20.
- M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.
Crossrefs
Programs
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GAP
P:=function(n) return Sum([1..n],x->Stirling2(n,x)*Factorial(x)); end;
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GAP
F:=function(n) return Sum([1..n],x->(-1)^(n-x)*Stirling1(n,x)*P(x)^2)/Factorial(n); end;
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Mathematica
m = 17; a670[n_] = Sum[ StirlingS2[n, k]*k!, {k, 0, n}]; Rest[ CoefficientList[ Series[ Sum[ a670[n]^2*(Log[1 + x]^n/n!), {n, 0, m}], {x, 0, m}], x]] (* Jean-François Alcover, Sep 02 2011, after g.f. *) Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#]]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *) -
PARI
{A000670(n)=sum(k=0,n,stirling(n, k,2)*k!)} {a(n)=polcoeff(sum(m=0,n,A000670(m)^2*log(1+x+x*O(x^n))^m/m!),n)} /* Paul D. Hanna, Nov 07 2009 */
Formula
a(n) = (Sum s(n, k) * P(k)^2)/n!, where P(n) is the number of labeled total preorders on {1, ..., n} (A000670), s are signed Stirling numbers of the first kind.
G.f.: Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^m. - Vladeta Jovovic, Mar 25 2006
Inverse binomial transform of A007322. - Vladeta Jovovic, Aug 17 2006
G.f.: Sum_{n>=0} 1/(2-(1+x)^n)/2^(n+1). - Vladeta Jovovic, Sep 23 2006
G.f.: Sum_{n>=0} A000670(n)^2*log(1+x)^n/n! where 1/(1-x) = Sum_{n>=0} A000670(n)*log(1+x)^n/n!. - Paul D. Hanna, Nov 07 2009
a(n) ~ n! / (2^(2+log(2)/2) * (log(2))^(2*(n+1))). - Vaclav Kotesovec, Dec 31 2013
Comments