A058790 -id:A058790 - cp's OEIS frontend

A050913 Pure 2-complexes on an infinite set of nodes with n multiple 2-simplexes. Also n-rowed binary matrices with all row sums 3, up to row and column permutation.

1, 1, 4, 16, 93, 652, 6369, 79568, 1256425, 24058631, 543204998, 14138916124, 417362929209, 13798729189578, 505990335048034, 20415765544541866, 900364519682003919, 43155049922002494115, 2236988329443856718604, 124862936181977439454012, 7476052709321753156375756, 478506183522725779096476581, 32638841238874891261354722405, 2365895836144423508306322639848, 181785988254681334224483607437510, 14771116583797935886529061991645404, 1266545494725474774697216198539818982

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation

Row n=3 of A331461.

Cf. A050911, A050912, A001400, A014395, A082789, A082790, A058790.

More terms from T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003

A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 3, 1, 1, 0, 1, 11, 12, 4, 1, 1, 0, 1, 26, 66, 25, 5, 1, 1, 0, 1, 68, 445, 278, 44, 6, 1, 1, 0, 1, 177, 4279, 5532, 966, 73, 7, 1, 1, 0, 1, 497, 53340, 200589, 53535, 2957, 112, 8, 1, 1, 0, 1, 1476, 846254, 11662671, 7043925, 431805, 8149, 166, 9, 1, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

An n-regular set multipartition is a finite multiset of nonempty sets in which each element appears in n blocks.
A set multipartition is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of nonequivalent binary matrices with k distinct columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.
A(n,k) is the number of non-isomorphic set-systems with k parts each of size n.

			Array begins:
===============================================
n\k | 0 1 2  3    4      5       6        7
----+------------------------------------------
  0 | 1 1 0  0    0      0       0        0 ...
  1 | 1 1 1  1    1      1       1        1 ...
  2 | 1 1 2  5   11     26      68      177 ...
  3 | 1 1 3 12   66    445    4279    53340 ...
  4 | 1 1 4 25  278   5532  200589 11662671 ...
  5 | 1 1 5 44  966  53535 7043925 ...
  6 | 1 1 6 73 2957 431805 ...
  ...
The A(2,3) = 5 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 1]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 1]
  [0 1 0]  [0 0 1]  [0 0 1]  [0 0 1]
  [0 0 1]  [0 0 1]
  [0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152 (first 17 antidiagonals)

Rows n=1..6 are A000012, A000664, A058790, A058791, A058792, A058793.

Cf. A283877, A306019, A330942, A331126, A331277, A331461, A331509, A331510.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))[k]}
T(n,k)={my(m=n*k, s=0); if(m==0, k<=1, forpart(q=m, my(g=sum(t=1, k, K(q, t, n)*x^t/t) + O(x*x^k)); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), k)); s/m!)}
{ for(n=0, 6, for(k=0, 5, print1(T(n, k), ", ")); print) } \\ Andrew Howroyd, Jan 16 2024

A306019(n) = Sum_{d|n} A(n/d, d).

A082789 Number of nonisomorphic configurations of n triples in Steiner triple systems.

Original entry on oeis.org

1, 2, 5, 16, 56, 282, 1865, 17100, 207697, 3180571

Offset: 1

T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003

A configuration is a set of triples (of points) where every pair of points occurs in at most one triple. (A Steiner triple system is a set of triples where every pair occurs exactly once; thus configurations are often called partial Steiner triple systems.) The triples are also called blocks.
A 'generator' is 'a configuration where every point occurs in at least two blocks'. The term refers to the work of Horak, Phillips, Wallis & Yucas, who show that the number of occurrences of a configuration in a Steiner triple system is expressible as a linear form in the numbers of occurrences of the generators.
If we relax the restriction on the number of times a pair of points can occur in a configuration -- so that a configuration is just any multi-set of triples - then we get A050913.
If we allow a configuration to be any *set* of triples -- i.e., configurations with multiple occurrences of blocks are not allowed, but more than one pair is allowed -- then we get A058790.

			The five configurations of 3 triples are
.
     *---*---*       *---*---*
     *---*---*       *---*---*
     *---*---*        \
                       *
                        \
          *       *      *
         / \     /
        *   *   *                *
       /     \ /                /|
      *       *                * |
                              /  |
     *---*---*---*---*       *   *
              \               \  |
               *               * |
                \               \|
                 *               *

Mike Grannell and Terry Griggs, 'Configurations in Steiner triple systems', in Combinatorial Designs and their Applications, Chapman & Hall, CRC Research Notes in Math. 403 (1999), 103-126.
Horak, P., Phillips, N. K. C., Wallis, W. D. and Yucas, J. L., Counting frequencies of configurations in Steiner triple systems. Ars Combin. 46 (1997), 65-75.

A. D. Forbes, M. J. Grannell and T. S. Griggs, Configurations and trades in Steiner triple systems, Australasian J. Combin. 29 (2004), 75-84.

Cf. A082790, A050913, A058790.

A058791 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 4 subsets of the cover and that intersection of every 4 subsets of the cover contains at most one point.

Original entry on oeis.org

1, 4, 25, 278, 5532, 200589, 11662671, 981458725, 111213730857, 16212510876591, 2947663429595884, 653299790376494253, 173373367472722467460, 54295204354760461199529, 19822683467350158062736540

Offset: 1

Vladeta Jovovic, Nov 30 2000

Cover may include multiple occurrences of a subset. Also n-rowed binary matrices with distinct rows and all row sums 4.

Vladeta Jovovic, Formula

Row n=4 of A331508.

Cf. A050535, A050913, A058783, A058784, A058785.

a(8)-a(12) from Andrew Howroyd, Jan 29 2020

a(13)-a(15) from Sean A. Irvine, Aug 26 2022

A058792 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 5 subsets of the cover and that intersection of every 5 subsets of the cover contains at most one point.

Original entry on oeis.org

1, 5, 44, 966, 53535, 7043925, 1775617861, 738134963014, 461550358993886, 409321620633426443, 493892363593918906465

Offset: 1

Vladeta Jovovic, Nov 30 2000

Cover may include multiple occurrences of a subset. Also n-rowed binary matrices with distinct rows and all row sums 5.

Vladeta Jovovic, Formula

Row n=5 of A331508.

Cf. A050535, A050913, A058783, A058784, A058785.

a(7)-a(10) from Andrew Howroyd, Jan 29 2020

a(11) from Sean A. Irvine, Aug 27 2022

A058793 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 6 subsets of the cover and that intersection of every 6 subsets of the cover contains at most one point.

Original entry on oeis.org

1, 6, 73, 2957, 431805, 193413963, 198914729182, 389238347659505, 1292594846197639190, 6753309741099683530616

Offset: 1

Vladeta Jovovic, Nov 30 2000

Cover may include multiple occurrences of a subset. Also n-rowed binary matrices with distinct rows and all row sums 6.

Vladeta Jovovic, Formula

Row n=6 of A331508.

Cf. A050535, A050913, A058783, A058784, A058785.

a(6)-a(10) from Andrew Howroyd, Jan 29 2020

cp's OEIS Frontend

A050913 Pure 2-complexes on an infinite set of nodes with n multiple 2-simplexes. Also n-rowed binary matrices with all row sums 3, up to row and column permutation.

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A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

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A082789 Number of nonisomorphic configurations of n triples in Steiner triple systems.

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A058791 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 4 subsets of the cover and that intersection of every 4 subsets of the cover contains at most one point.

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A058792 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 5 subsets of the cover and that intersection of every 5 subsets of the cover contains at most one point.

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A058793 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 6 subsets of the cover and that intersection of every 6 subsets of the cover contains at most one point.

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Author

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