A050913 -id:A050913 - cp's OEIS frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017
Row sums of the triangle defined by the Multiset Transformation of A076864,
;
1;
2 1;
5 2 1;
12 8 2 1;
33 22 8 2 1;
103 72 26 8 2 1;
333 229 87 26 8 2 1;
1183 782 295 92 26 8 2 1;
4442 2760 1036 315 92 26 8 2 1;
17576 10270 3735 1129 321 92 26 8 2 1;
72810 39770 13976 4117 1154 321 92 26 8 2 1;
314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
- R. J. Mathar, Jul 18 2017
Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..50
George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, Permutation invariant Gaussian matrix models for financial correlation matrices, arXiv:2306.04569 [q-fin.ST], 2023.
Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation
Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.

Cf. A001399, A003082, A014395, A014396, A014397, A014398.
Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017

Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

More terms from Sean A. Irvine, Oct 02 2011

A331461 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 8, 4, 1, 1, 1, 7, 23, 16, 5, 1, 1, 1, 11, 66, 93, 30, 6, 1, 1, 1, 15, 212, 652, 332, 50, 7, 1, 1, 1, 22, 686, 6369, 6414, 1062, 80, 8, 1, 1, 1, 30, 2389, 79568, 226041, 56712, 3117, 120, 9, 1, 1, 1, 42, 8682, 1256425, 12848128, 7295812, 441881, 8399, 175, 10, 1, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

A(n,k) is the number of non-isomorphic set multipartitions (multiset of sets) with k parts each part has size n.

			Array begins:
===========================================================
n\k | 0 1 2   3    4       5          6              7
----+-----------------------------------------------------
  0 | 1 1 1   1    1       1          1              1 ...
  1 | 1 1 2   3    5       7         11             15 ...
  2 | 1 1 3   8   23      66        212            686 ...
  3 | 1 1 4  16   93     652       6369          79568 ...
  4 | 1 1 5  30  332    6414     226041       12848128 ...
  5 | 1 1 6  50 1062   56712    7295812     1817321457 ...
  6 | 1 1 7  80 3117  441881  195486906   200065951078 ...
  7 | 1 1 8 120 8399 3006771 4298181107 17131523059493 ...
  ...
The A(2,3) = 8 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 1]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 1 0]  [1 0 1]  [1 1 0]  [1 0 1]  [1 1 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 0 1]  [0 1 0]  [0 0 1]  [0 1 1]
  [0 1 0]  [0 0 1]  [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

Rows n=0..6 are A000012, A000041, A050535, A050913, A058783, A058784, A058785.
Columns k=0..4 are A000012, A000012, A000027(n+1), A002624, A331720.
Cf. A188392, A262809, A304942, A306018, A330942, A331485, A331508, A331509, A331510.

\\ See A304942 for Blocks
T(n,k)={Blocks(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n,k), ", ")); print) }

A306018(n) = Sum_{d|n} A(n/d, d).

A058783 Number of n-rowed binary matrices with all row sums 4, up to row and column permutation; bipartite graphs with all nodes of degree 4 in a distinguished bipartite n-block, up to isomorphism.

Original entry on oeis.org

1, 1, 5, 30, 332, 6414, 226041, 12848128, 1064292052, 119252205304, 17239075745190, 3113843614322530, 686491853010870156, 181386885029173595218, 56595769613704915215101, 20597536264356706999502705, 8652615103513713632218678155, 4157673185369121151936091009448

Offset: 0

Vladeta Jovovic, Nov 28 2000

nonn

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

Row n=4 of A331461.

Cf. A050535, A050913, A058784, A058785.

Terms a(8)-a(17) from Max Alekseyev, May 04 2018

A058784 Number of n-rowed binary matrices with all row sums 5, up to row and column permutation; bipartite graphs with all nodes of degree 5 in a distinguished bipartite n-block, up to isomorphism.

Original entry on oeis.org

1, 1, 6, 50, 1062, 56712, 7295812, 1817321457, 750572034647, 467477187494249, 413492386704135759, 498000976932085045800, 791133947048635010571251, 1616618616930983136958492360, 4162734189568898641317813946712, 13276910075946265356264268787123401

Offset: 0

Vladeta Jovovic, Nov 28 2000

nonn

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

Row n=5 of A331461.

Cf. A050535, A050913, A058783, A058785.

a(7)-a(15) from Max Alekseyev, May 04 2018

A058785 Number of n-rowed binary matrices with all row sums 6, up to row and column permutation; bipartite graphs with all nodes of degree 6 in a distinguished bipartite n-block, up to isomorphism.

Original entry on oeis.org

1, 1, 7, 80, 3117, 441881, 195486906, 200065951078, 390629444879796, 1295710020278986959, 6764950163433890601997, 52637015698577710285832949, 585529138290221945767868247037, 9010635399744213671095287436920755

Offset: 0

Vladeta Jovovic, Nov 28 2000

nonn

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

Row n=6 of A331461.

Cf. A050535, A050913, A058783, A058784.

a(6)-a(13) from Max Alekseyev, May 04 2018

A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 3, 14, 44, 129, 316, 714, 1452, 2775, 4963, 8478, 13838, 21827, 33306, 49504, 71754, 101871, 141807, 194128, 261570, 347633, 456026, 591384, 758596, 963657, 1212861, 1513806, 1874440, 2304225, 2813030, 3412466, 4114608, 4933519

Offset: 0

Vladeta Jovovic, Nov 24 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000
V. Jovovic, Illustration of initial terms
V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 3 of A318951.
Cf. A002817, A052282, A058390, A058391, A058392.
Cf. A001501, A050535, A050913, A058528, A058783, A058784, A058785.

a[n_] := (m = Mod[n, 6]; (n^3 + 9*n^2 + 39*n + 120)*n^3 + Which[m == 0, 12*(23*n^2 + 32*n + 24), m == 1 || m == 5, 249*n^2 + 303*n + 143, m == 2 || m == 4, 4*(69*n^2 + 96*n + 56), m == 3, 3*(83*n^2 + 101*n + 69)])/288; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 12 2011, after Vladeta Jovovic *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(3, 3, n); \\ Andrew Howroyd, Sep 05 2018

a(n) = (1/6)*(C(C(n + 2, 2) + 2, 3) + 3/2*floor((n + 2)/2)*(C(n + 2, 2) - floor((n + 2)/2)) + 3*C(floor((n + 2)/2) + 2, 3) + 2*floor(C(n + 2, 2)/3) + 2*C(C(n + 2, 2) - 3*floor(C(n + 2, 2)/3) + 2, 3)).

Empirical G.f.: -(x^8 + 3*x^7 + 14*x^6 + 12*x^5 + 15*x^4 + 9*x^3 + 5*x^2 + 1) / ((x-1)^7*(x+1)^3*(x^2+x+1)). - Colin Barker, Dec 27 2012

More terms from Marc LeBrun, Dec 11 2000

A058790 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 3 subsets of the cover and that intersection of every 3 subsets of the cover contains at most one point.

Original entry on oeis.org

1, 3, 12, 66, 445, 4279, 53340, 846254, 16333946, 371976963, 9763321109, 290473143807, 9674133467729, 357177322891321, 14503958827502886, 643502334799711633, 31018731336031551119, 1616523352051185316626, 90689288905913623412837, 5456178840303106057314759, 350830170593891706361540379

Offset: 1

Vladeta Jovovic, Nov 30 2000

nonn

Cover may include multiple occurrences of a subset. Also n-rowed binary matrices with distinct rows and all row sums 3.

For labeled case see Goulden I. P., Jackson D. M., Combinatorial Enumeration, John Wiley and Sons, New York, 1983.

Vladeta Jovovic, Formula

Row n=3 of A331508.

Cf. A050535, A050913, A058783, A058784, A058785, A082789, A082790.

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

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.

A305027 Array read by antidiagonals: T(n,m) is the number of nonisomorphic binary n X m matrices with 3 1's per column under row and column permutations (m >= 3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 7, 5, 1, 1, 1, 4, 11, 17, 6, 1, 1, 1, 4, 14, 40, 35, 9, 1, 1, 1, 4, 15, 62, 122, 76, 11, 1, 1, 1, 4, 16, 78, 272, 410, 149, 15, 1, 1, 1, 4, 16, 87, 427, 1307, 1270, 291, 18, 1, 1, 1, 4, 16, 91, 544, 2754, 6178, 3888, 539, 23, 1

Offset: 0

Andrew Howroyd, May 24 2018

Also, the number of pure 2-complexes on m nodes with n multiple 2-simplexes.

			Array begins:
========================================================
n\m| 3  4   5    6     7      8      9     10     11
---+----------------------------------------------------
0  | 1  1   1    1     1      1      1      1      1 ...
1  | 1  1   1    1     1      1      1      1      1 ...
2  | 1  2   3    4     4      4      4      4      4 ...
3  | 1  3   7   11    14     15     16     16     16 ...
4  | 1  5  17   40    62     78     87     91     92 ...
5  | 1  6  35  122   272    427    544    606    635 ...
6  | 1  9  76  410  1307   2754   4251   5343   5939 ...
7  | 1 11 149 1270  6178  18247  36455  54621  67609 ...
8  | 1 15 291 3888 28687 122038 327774 616020 891831 ...
...

StackExchange, How many arrays with crossed cells, order of rows/columns irrelevant, Dec 13 2013

Columns m=4..7 are A001400, A014395, A050911, A050912.
A diagonal is A247596.
Cf. A050913 (infinite m), A304942.

\\ See A304942 for Blocks
for(n=1, 8, for(m=3, 11, print1(Blocks(n, m, 3), ", ")); print)

A082790 Number of nonisomorphic configurations of degree >= 2 (or generators) of n triples in Steiner triple systems.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 19, 153, 1615, 25180, 479238, 10695820

Offset: 1

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

nonn

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 you 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.

Forbes, Grannell & Griggs, 'Configurations and trades in Steiner triple systems', in preparation.
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.

Cf. A082789, A050913.

cp's OEIS Frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

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A331461 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.

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A058783 Number of n-rowed binary matrices with all row sums 4, up to row and column permutation; bipartite graphs with all nodes of degree 4 in a distinguished bipartite n-block, up to isomorphism.

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A058784 Number of n-rowed binary matrices with all row sums 5, up to row and column permutation; bipartite graphs with all nodes of degree 5 in a distinguished bipartite n-block, up to isomorphism.

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A058785 Number of n-rowed binary matrices with all row sums 6, up to row and column permutation; bipartite graphs with all nodes of degree 6 in a distinguished bipartite n-block, up to isomorphism.

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A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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A058790 Number of covers of an unlabeled n-set such that every point of the set is covered by exactly 3 subsets of the cover and that intersection of every 3 subsets of the cover contains at most one point.

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

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A305027 Array read by antidiagonals: T(n,m) is the number of nonisomorphic binary n X m matrices with 3 1's per column under row and column permutations (m >= 3).

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