A319781 -id:A319781 - cp's OEIS frontend

A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

1, 0, 0, 0, 0, 0, 1, 1, 3, 8, 18

Offset: 0

Gus Wiseman, Sep 27 2018

A set system is a finite set of finite nonempty sets. It is intersecting if no two parts are disjoint. The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 1 through a(9) = 8 set systems:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317752, A317755, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319763, A319765, A319779, A319781.

A319775 Number of non-isomorphic multiset partitions of weight n with empty intersection and no part containing all the vertices.

Original entry on oeis.org

1, 0, 1, 4, 16, 52, 185, 625, 2226, 7840, 28405

Offset: 0

Gus Wiseman, Sep 27 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 16 multiset partitions:
2: {{1},{2}}
3: {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{2}}
   {{1},{2},{3}}
4: {{1},{2,2,2}}
   {{1},{2,3,3}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{1},{2,2}}
   {{1},{1},{2,3}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{1},{2},{2}}
   {{1},{2},{2},{2}}
   {{1},{2},{3},{3}}
   {{1},{2},{3},{4}}

Cf. A007716, A281116, A283877, A316980, A317752, A317755, A317757.

Cf. A319778, A319779, A319781, A319783.

A319783 Number of set systems spanning n vertices with empty intersection whose dual is also a set system with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 203, 490572

Offset: 0

Gus Wiseman, Sep 27 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

			The a(3) = 1 set system is {{1,2},{1,3},{2,3}}.

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757.

Cf. A319775, A319778, A319779, A319781.

A319751 Number of non-isomorphic set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 2, 6, 13, 35, 83, 217, 556, 1504, 4103, 11715, 34137, 103155, 320217, 1025757, 3376889, 11436712, 39758152, 141817521, 518322115, 1939518461, 7422543892, 29028055198, 115908161428, 472185530376, 1961087909565, 8298093611774, 35750704171225, 156734314212418

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

A set system is a finite set of finite nonempty sets. Its weight is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 set systems:
2: {{1},{2}}
3: {{1},{2,3}}
   {{1},{2},{3}}
4: {{1},{2,3,4}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}
5: {{1},{2,3,4,5}}
   {{1,2},{3,4,5}}
   {{1},{2},{3,4,5}}
   {{1},{4},{2,3,4}}
   {{1},{2,3},{4,5}}
   {{1},{2,4},{3,4}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{4},{1,2},{3,4}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{3},{4,5}}
   {{1},{2},{4},{3,4}}
   {{1},{2},{3},{4},{5}}

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

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319748, A319755, A319778, A319781, A319791.

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, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q,n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t]-subst(u[t],x,x^2), O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t] - subst(x^t*u[t],x,x^2), O(x*x^n)))*(1+x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A319792 Number of non-isomorphic connected set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 9, 22, 69, 190, 567, 1640, 5025, 15404, 49048, 159074, 531165, 1813627, 6352739, 22759620, 83443086, 312612543, 1196356133, 4672620842, 18615188819, 75593464871, 312729620542, 1317267618429, 5646454341658, 24618309943464, 109123789229297

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(6) = 9 connected set systems:
4:   {{1},{2},{1,2}}
5:  {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
6: {{1},{1,4},{2,3,4}}
   {{1},{2,3},{1,2,3}}
   {{3},{4},{1,2,3,4}}
   {{3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,4},{3,4}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}

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

Cf. A007716, A007718, A049311, A056156, A281116, A283877, A300913, A305854, A316980, A317752, A317755, A317757.

Cf. A319077, A319748, A319751, A319755, A319778, A319781, A319790.

a(n) = A300913(n) - A283877(n) + A319751(n). - Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

A319793 Number of non-isomorphic connected strict multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 24, 96, 412, 1607, 6348, 24580, 96334, 378569, 1508220, 6079720, 24879878, 103335386, 436032901, 1869019800, 8139613977, 36008825317, 161794412893, 738167013847, 3418757243139, 16068569129711, 76622168743677, 370571105669576, 1817199912384794

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(5) = 4 multiset partitions:
4:  {{1},{2},{1,2}}
5: {{1},{2},{1,2,2}}
   {{1},{1,2},{2,2}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}

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

Cf. A007716, A007718, A056156, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319748, A319755, A319778, A319781, A319791.

a(n) = A319557(n) - A316980(n) + A319077(n). - Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

cp's OEIS Frontend

A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

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A319775 Number of non-isomorphic multiset partitions of weight n with empty intersection and no part containing all the vertices.

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A319783 Number of set systems spanning n vertices with empty intersection whose dual is also a set system with empty intersection.

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A319751 Number of non-isomorphic set systems of weight n with empty intersection.

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A319792 Number of non-isomorphic connected set systems of weight n with empty intersection.

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A319793 Number of non-isomorphic connected strict multiset partitions (sets of multisets) of weight n with empty intersection.

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