A306008 -id:A306008 - cp's OEIS frontend

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

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

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

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.

A305999 Number of unlabeled spanning intersecting set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 6, 76, 12916

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. S is spanning if every vertex is contained in some edge. A singleton is an edge containing only one vertex.

			Non-isomorphic representative of the a(3) = 6 set-systems:
{{1,2,3}}
{{1,3},{2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A051185, A048143, A261006, A058891, A261005, A304998, A305854-A305857, A305935, A306000, A306001, A306008.

a(n) = A306001(n) - A306001(n-1) for n > 0. - Andrew Howroyd, Aug 12 2019

a(5) from Andrew Howroyd, Aug 12 2019

A305935 Number of labeled spanning intersecting set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 12, 809, 1146800, 899927167353, 291136684655893185321964, 14704020783497694096988185391720223222562121969, 12553242487939982849962414795232892198542733492886483991398790450208264017757788101836749760

Offset: 0

Gus Wiseman, Jun 15 2018

nonn

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. S is spanning if every vertex is contained in some edge. A singleton is an edge containing only one vertex.

			The a(3) = 12 spanning intersecting set-systems with no singletons:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{1,2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A006126, A051185, A048143, A058891, A305001, A305843, A305844, A305854-A305857, A305999, A306000, A306001, A306008.

a(n) = A305843(n) - n * A003465(n-1).

Inverse binomial transform of A306000. - Andrew Howroyd, Aug 12 2019

a(6)-a(8) from Giovanni Resta, Jun 20 2018

a(9) from Andrew Howroyd, Aug 12 2019

cp's OEIS Frontend

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

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

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A305999 Number of unlabeled spanning intersecting set-systems on n vertices with no singletons.

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A305935 Number of labeled spanning intersecting set-systems on n vertices with no singletons.

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