A116539 - OEIS

%I #26 Feb 25 2020 08:14:09

%S 1,1,2,7,28,134,729,4408,29256,210710,1633107,13528646,119117240,

%T 1109528752,10889570768,112226155225,1210829041710,13640416024410,

%U 160069458445202,1952602490538038,24712910192430620,323964329622503527,4391974577299578248,61488854148194151940

%N Number of zero-one matrices with n ones and no zero rows or columns and with distinct rows, up to permutation of rows.

%C Also the number of labeled hypergraphs spanning an initial interval of positive integers with edge-sizes summing to n. - _Gus Wiseman_, Dec 18 2018

%H Alois P. Heinz, <a href="/A116539/b116539.txt">Table of n, a(n) for n = 0..300</a>

%H P. J. Cameron, T. Prellberg and D. Stark, <a href="https://arxiv.org/abs/math/0510155">Asymptotics for incidence matrix classes </a>, arXiv:math/0510155 [math.CO], 2005-2006.

%H M. Klazar, <a href="https://arxiv.org/abs/math/0305048">Extremal problems for ordered hypergraphs</a>, arXiv:math/0305048 [math.CO], 2003.

%e From _Gus Wiseman_, Dec 18 2018: (Start)

%e The a(3) = 7 edge-sets:

%e     {{1,2,3}}

%e    {{1},{1,2}}

%e    {{2},{1,2}}

%e    {{1},{2,3}}

%e    {{2},{1,3}}

%e    {{3},{1,2}}

%e   {{1},{2},{3}}

%e Inequivalent representatives of the a(4) = 28 0-1 matrices:

%e   [1111]

%e .

%e   [100][1000][010][0100][001][0010][0001][110][110][1100][101][1010][1001]

%e   [111][0111][111][1011][111][1101][1110][101][011][0011][011][0101][0110]

%e .

%e   [10][100][100][1000][100][100][1000][1000][010][010][0100][0100][0010]

%e   [01][010][010][0100][001][001][0010][0001][001][001][0010][0001][0001]

%e   [11][101][011][0011][110][011][0101][0110][110][101][1001][1010][1100]

%e .

%e   [1000]

%e   [0100]

%e   [0010]

%e   [0001]

%e (End)

%p b:= proc(n, i, k) b(n, i, k):=`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j,

%p       min(n-i*j, i-1), k)*binomial(binomial(k, i), j), j=0..n/i)))

%p     end:

%p a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 13 2019

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k]*Binomial[Binomial[k, i], j], {j, 0, n/i}]]];

%t a[n_] := Sum[Sum[b[n, n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}], {k, 0, n}];

%t a /@ Range[0, 23] (* _Jean-François Alcover_, Feb 25 2020, after _Alois P. Heinz_ *)

%Y Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

%Y Cf. A049311, A058891, A101370, A255906, A283877, A306021, A319190.

%Y Row sums of A326914 and of A326962.

%K nonn

%O 0,3

%A _Vladeta Jovovic_, Mar 27 2006

%E a(0)=1 prepended and more terms added by _Alois P. Heinz_, Sep 13 2019