A293607 -id:A293607 - cp's OEIS frontend

A300913 Number of non-isomorphic connected set-systems of weight n.

1, 1, 1, 2, 4, 7, 18, 37, 96, 239, 658, 1810, 5358, 16057, 50373, 161811, 536964, 1826151, 6380481, 22822280, 83587920, 312954111, 1197178941, 4674642341, 18620255306, 75606404857, 312763294254, 1317356836235, 5646694922172, 24618969819915, 109125629486233, 491554330852608

Offset: 0

Gus Wiseman, Jun 19 2018

nonn

The weight of a set-system is the sum of cardinalities of the sets. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 set systems:
1: {{1}}
2: {{1,2}}
3: {{1,2,3}}
   {{2},{1,2}}
4: {{1,2,3,4}}
   {{3},{1,2,3}}
   {{1,3},{2,3}}
   {{1},{2},{1,2}}
5: {{1,2,3,4,5}}
   {{4},{1,2,3,4}}
   {{1,4},{2,3,4}}
   {{2,3},{1,2,3}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{3},{1,3},{2,3}}
Non-isomorphic representatives of the a(6) = 18 connected set-systems:
  {{1,2,3,4,5,6}}
  {{5},{1,2,3,4,5}}
  {{1,5},{2,3,4,5}}
  {{3,4},{1,2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,4}}
  {{1},{1,4},{2,3,4}}
  {{1},{2,3},{1,2,3}}
  {{3},{4},{1,2,3,4}}
  {{3},{1,4},{2,3,4}}
  {{3},{2,3},{1,2,3}}
  {{4},{1,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}

Jean-François Alcover, Table of n, a(n) for n = 0..50 [using Andrew Howroyd's b-file for A283877]

Cf. A007716, A055621, A283877, A293606, A293607.

A283877 = Import["https://oeis.org/A283877/b283877.txt", "Table"][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[A283877 // Rest] (* Jean-François Alcover, Nov 07 2019, updated Mar 17 2020 *)

Inverse Euler transform of A283877.

a(11)-a(31) from Jean-François Alcover, Nov 07 2019

A293606 Number of unlabeled antichains of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 20, 33, 72, 139

Offset: 0

Gus Wiseman, Oct 13 2017

An antichain is a finite set of finite nonempty sets, none of which is a subset of any other. The weight of an antichain is the sum of cardinalities of its elements.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n where every vertex is the unique common element of some subset of the edges. For example, the a(1) = 1 through a(6) = 20 set multipartitions are:
  {1}  {1}{1}  {1}{1}{1}  {1}{2}{12}    {1}{2}{2}{12}    {12}{13}{23}
       {1}{2}  {1}{2}{2}  {1}{1}{1}{1}  {1}{2}{3}{23}    {1}{2}{12}{12}
               {1}{2}{3}  {1}{1}{2}{2}  {1}{1}{1}{1}{1}  {1}{2}{13}{23}
                          {1}{2}{2}{2}  {1}{1}{2}{2}{2}  {1}{2}{3}{123}
                          {1}{2}{3}{3}  {1}{2}{2}{2}{2}  {1}{1}{2}{2}{12}
                          {1}{2}{3}{4}  {1}{2}{2}{3}{3}  {1}{1}{2}{3}{23}
                                        {1}{2}{3}{3}{3}  {1}{2}{2}{2}{12}
                                        {1}{2}{3}{4}{4}  {1}{2}{3}{3}{23}
                                        {1}{2}{3}{4}{5}  {1}{2}{3}{4}{34}
                                                         {1}{1}{1}{1}{1}{1}
                                                         {1}{1}{1}{2}{2}{2}
                                                         {1}{1}{2}{2}{2}{2}
                                                         {1}{1}{2}{2}{3}{3}
                                                         {1}{2}{2}{2}{2}{2}
                                                         {1}{2}{2}{3}{3}{3}
                                                         {1}{2}{3}{3}{3}{3}
                                                         {1}{2}{3}{3}{4}{4}
                                                         {1}{2}{3}{4}{4}{4}
                                                         {1}{2}{3}{4}{5}{5}
                                                         {1}{2}{3}{4}{5}{6}
(End)

			Non-isomorphic representatives of the a(5) = 9 antichains are:
((12345)),
((1)(2345)), ((12)(134)), ((12)(345)),
((1)(2)(345)), ((1)(23)(45)), ((2)(13)(14)),
((1)(2)(3)(45)),
((1)(2)(3)(4)(5)).

Cf. A006126, A006602, A007411, A007716, A048143, A049311, A283877, A293607.

Cf. A000372, A000612, A003182, A014466, A055621, A293993, A326704, A326972.

Euler transform of A293607.

A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 12, 23, 84, 204, 682, 1977, 6546, 21003, 72038, 248055, 888771, 3240578, 12152775, 46527471, 182339441, 729405164, 2979121279, 12407308136, 52670355242, 227725915268, 1002285274515, 4487915293698, 20434064295155, 94559526596293, 444527730210294, 2122005930659752

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. A singleton is a multiset of size 1. The weight of a multiset partition is the sum of sizes of its elements. Weight is generally not the same as number of vertices.
Also non-isomorphic multiset partitions of weight n with no endpoints, where an endpoint is a vertex appearing only once (degree 1). For example, non-isomorphic representations of the a(4) = 12 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}

			The a(4) = 12 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The set-system version is A330054 (no endpoints) or A306005 (no singletons).
Non-isomorphic multiset partitions are A007716.
Set-systems with no singletons are A016031.
Cf. A049311, A283877, A293606, A293607, A306008, A317533, A317794, A317795, A320665, A330053, A330055, A330058.

\\ compare with similar program for A007716.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, 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)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t)) + O(x*x^k), -k)}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 15 2023

Extended by Gus Wiseman, Dec 09 2019

Terms a(11) and beyond from Andrew Howroyd, Jan 15 2023

A306005 Number of non-isomorphic set-systems of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 19, 51, 106, 274, 647, 1773, 4664, 13418, 38861, 118690, 370588, 1202924, 4006557, 13764760, 48517672, 175603676, 651026060, 2471150365, 9590103580, 38023295735, 153871104726, 635078474978, 2671365285303, 11444367926725, 49903627379427

Offset: 0

Gus Wiseman, Jun 16 2018

nonn

A set-system is a finite set of finite nonempty sets (edges). The weight is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 12 set-systems:
  {{1,2,3,4,5,6}}
  {{1,2},{3,4,5,6}}
  {{1,5},{2,3,4,5}}
  {{3,4},{1,2,3,4}}
  {{1,2,3},{4,5,6}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{3,4},{5,6}}
  {{1,2},{3,5},{4,5}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The complement is counted by A330053.

Cf. A007716, A034691, A048143, A049311, A054921, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

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)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

a(n) = A283877(n) - A330053(n). - Gus Wiseman, Dec 09 2019

Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

A293994 Number of unlabeled multiset clutters of weight n.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 39, 88, 265

Offset: 0

Gus Wiseman, Oct 21 2017

A multiset clutter is a connected antichain of finite multisets. The weight of a multiset clutter is the sum of cardinalities (counting multiplicity) of its edges.

			Non-isomorphic representatives of the a(5) = 13 multiset clutters are:
((11111)), ((11112)), ((11122)), ((11123)), ((11223)), ((11234)), ((12345)), ((11)(122)), ((11)(123)), ((12)(111)), ((12)(113)), ((12)(134)), ((13)(122)).

Cf. A007716, A048143, A286520, A293607, A293993.

A306006 Number of non-isomorphic intersecting set-systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 10, 16, 30, 57, 109, 209, 431, 873, 1850, 3979, 8819, 19863

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. The weight of S is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 10 set-systems:
{{1,2,3,4,5,6}}
{{5},{1,2,3,4,5}}
{{1,5},{2,3,4,5}}
{{3,4},{1,2,3,4}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{3},{2,3},{1,2,3}}
{{4},{1,4},{2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,4},{2,4},{3,4}}

Cf. A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

a(10)-a(17) from Bert Dobbelaere, May 04 2025

A304887 Number of non-isomorphic blobs of weight n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 8, 14

Offset: 0

Gus Wiseman, May 20 2018

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch. The weight of a blob is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A275307).

			Non-isomorphic representatives of the a(8) = 8 blobs are the following:
  {{1,2,3,4,5,6,7,8}}
  {{1,5,6},{2,3,4,5,6}}
  {{1,2,5,6},{3,4,5,6}}
  {{1,3,4,5},{2,3,4,5}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,4},{1,5},{2,3,4,5}}
  {{2,4},{1,2,5},{3,4,5}}
  {{1,2},{1,3},{2,4},{3,4}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A030019, A035053, A048143, A275307, A283877, A286520, A293607, A293993, A293994, A304118, A304867.

A306007 Number of non-isomorphic intersecting antichains of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 14, 22

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting antichain S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection, and none of which is a subset of any other. The weight of S is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(8) = 14 set-systems:
{{1,2,3,4,5,6,7,8}}
{{1,7},{2,3,4,5,6,7}}
{{1,2,7},{3,4,5,6,7}}
{{1,5,6},{2,3,4,5,6}}
{{1,2,3,7},{4,5,6,7}}
{{1,2,5,6},{3,4,5,6}}
{{1,3,4,5},{2,3,4,5}}
{{1,2},{1,3,4},{2,3,4}}
{{1,4},{1,5},{2,3,4,5}}
{{1,5},{2,4,5},{3,4,5}}
{{1,6},{2,6},{3,4,5,6}}
{{1,6},{2,3,6},{4,5,6}}
{{2,4},{1,2,5},{3,4,5}}
{{1,5},{2,5},{3,5},{4,5}}

Cf. A007411, A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

A306008 Number of non-isomorphic intersecting set-systems of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 7, 10, 21, 39, 78

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 7 set-systems:
{{1,2,3,4,5,6}}
{{1,5},{2,3,4,5}}
{{3,4},{1,2,3,4}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,4},{2,4},{3,4}}

Cf. A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306007.

A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

Original entry on oeis.org

1, 0, 1, 2, 10, 128

Offset: 0

Gus Wiseman, May 24 2018

nonn

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch.

			Non-isomorphic representatives of the a(4) = 10 blobs:
  {{1,2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A030019, A035053, A048143, A054921, A134957, A144959, A275307, A286520, A293607, A304118, A304717, A304867.

cp's OEIS Frontend

A300913 Number of non-isomorphic connected set-systems of weight n.

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A293606 Number of unlabeled antichains of weight n.

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A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

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A306005 Number of non-isomorphic set-systems of weight n with no singletons.

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PARI

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A293994 Number of unlabeled multiset clutters of weight n.

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A306006 Number of non-isomorphic intersecting set-systems of weight n.

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A304887 Number of non-isomorphic blobs of weight n.

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A306007 Number of non-isomorphic intersecting antichains of weight n.

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A306008 Number of non-isomorphic intersecting set-systems of weight n with no singletons.

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