A316980 -id:A316980 - cp's OEIS frontend

A319639 Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

1, 1, 1, 2, 20, 2043

Offset: 0

Gus Wiseman, Sep 25 2018

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

			The a(1) = 1 through a(3) = 2 antichain covers:
1: {{1}}
2: {{1},{2}}
3: {{1},{2},{3}}
   {{1,2},{1,3},{2,3}}

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

A319748 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 10, 25, 72, 182, 502, 1332, 3720, 10380, 30142, 88842, 270569, 842957, 2703060, 8885029, 29990388, 103743388, 367811233, 1334925589, 4957151327, 18817501736, 72972267232, 288863499000, 1166486601571, 4802115258807, 20141268290050, 86017885573548, 373852868791639

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a set multipartition 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) = 10 set multipartitions:
  {{1},{2}}   {{1},{2,3}}     {{1},{2,3,4}}
             {{1},{2},{2}}    {{1,2},{3,4}}
             {{1},{2},{3}}   {{1},{1},{2,3}}
                             {{1},{2},{1,2}}
                             {{1},{2},{3,4}}
                             {{1},{3},{2,3}}
                            {{1},{1},{2},{2}}
                            {{1},{2},{2},{2}}
                            {{1},{2},{3},{3}}
                            {{1},{2},{3},{4}}

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

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

Cf. A319077, A319751, 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], O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t], 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

A319767 Number of non-isomorphic intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 1, 5, 73

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}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
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(1) = 1 through a(3) = 5 multiset partitions:
1: {{1}}
2: {{2},{1,2}}
3: {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319765, A319766, A319768, A319769, A319773, A319774.

A319786 Number of factorizations of n where no two factors are relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 12, 1, 2, 2, 4, 1, 1, 1, 4, 1

Offset: 1

Gus Wiseman, Sep 27 2018

nonn

First differs from A305193 at a(36) = 4, A305193(36) = 5.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018

			The a(48) = 7 factorizations are (2*2*2*6), (2*2*12), (2*4*6), (2*24), (4*12), (6*8), (48).

Antti Karttunen, Table of n, a(n) for n = 1..11520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A007716, A045778, A049311, A162247, A281116, A283877, A305193, A305854, A306006, A316980, A318749.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!Or@@CoprimeQ@@@Subsets[#,{2}]&]],{n,100}]

A319786(n, m=n, facs=List([])) = if(1==n, (1!=gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A319786(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Nov 07 2018

More terms from Antti Karttunen, Nov 07 2018

A319787 Number of intersecting multiset partitions of normal multisets of size n.

Original entry on oeis.org

1, 1, 3, 8, 27, 95, 373, 1532, 6724

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset is normal if it spans an initial interval of positive integers.

A multiset partition is intersecting iff no two parts are disjoint.

			The a(1) = 1 through a(3) = 8 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,1,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
   {{1},{1,2}}
   {{1},{1},{1}}

Cf. A007716, A049311, A283877, A305854, A306006, A316980, A317752.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319784, A319789.

A323788 Number of non-isomorphic weight-n sets of multisets of multisets.

Original entry on oeis.org

1, 1, 5, 19, 88, 391, 1995, 10281

Offset: 0

Gus Wiseman, Jan 27 2019

Also the number of non-isomorphic strict multiset partitions of multiset partitions of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 19 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{1}}    {{123}}
         {{1}{2}}    {{1}{11}}
         {{1}}{{2}}  {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}{1}{1}}
                     {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{2}}{{3}}

Cf. A005121, A007716, A049311, A050343, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.

Cf. A323787, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A323789 Number of non-isomorphic weight-n sets of sets of multisets.

Original entry on oeis.org

1, 1, 4, 15, 64, 269, 1310, 6460

Offset: 0

Gus Wiseman, Jan 27 2019

Also the number of non-isomorphic strict multiset partitions, with strict parts, of multiset partitions of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 15 multiset partition partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{2}}  {{1}{11}}
                     {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Cf. A007716, A049311, A050343, A283877, A316980, A317791, A318564, A318565, A318566, A318812.

Cf. A323787, A323788, A323790, A323791, A323792, A323793, A323794, A323795.

A323791 Number of non-isomorphic weight-n sets of multisets of sets.

Original entry on oeis.org

1, 1, 4, 13, 52, 196, 877, 3917

Offset: 0

Gus Wiseman, Jan 27 2019

All sets and multisets must be finite, and only the outermost may be empty.

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 13 sets of multisets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{1}}    {{1}{12}}
         {{1}{2}}    {{1}{23}}
         {{1}}{{2}}  {{1}{1}{1}}
                     {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{2}}{{3}}

Cf. A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.

Cf. A323787, A323788, A323789, A323790, A323792, A323793, A323794, A323795.

A323792 Number of non-isomorphic weight-n multisets of sets of sets.

Original entry on oeis.org

1, 1, 4, 11, 43, 145, 614, 2549

Offset: 0

Gus Wiseman, Jan 27 2019

All sets and multisets must be finite, and only the outermost may be empty.

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 11 multiset partitions:
  {{1}}  {{12}}      {{123}}
         {{1}{2}}    {{1}{12}}
         {{1}}{{1}}  {{1}{23}}
         {{1}}{{2}}  {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{1}}{{1}}
                     {{1}}{{1}}{{2}}
                     {{1}}{{2}}{{3}}

Cf. A007716, A049311, A050326, A050343, A255906, A283877, A306186, A316980, A318564, A318565, A318566.

Cf. A323787, A323788, A323789, A323790, A323791, A323793, A323794, A323795.

A323793 Number of non-isomorphic weight-n multisets of multisets of sets.

Original entry on oeis.org

1, 1, 5, 15, 65, 240, 1090, 4845

Offset: 0

Gus Wiseman, Jan 27 2019

Also the number of non-isomorphic multiset partitions of set multipartitions of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 15 multiset partitions:
  {{1}}  {{12}}      {{123}}
         {{1}{1}}    {{1}{12}}
         {{1}{2}}    {{1}{23}}
         {{1}}{{1}}  {{1}{1}{1}}
         {{1}}{{2}}  {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{1}}{{1}}
                     {{1}}{{1}}{{2}}
                     {{1}}{{2}}{{3}}

Cf. A007716, A049311, A283877, A306186, A316980, A318565, A318566.

Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323794, A323795.

cp's OEIS Frontend

A319639 Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

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A319748 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.

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A319767 Number of non-isomorphic intersecting set systems spanning n vertices whose dual is also an intersecting set system.

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A319786 Number of factorizations of n where no two factors are relatively prime.

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A319787 Number of intersecting multiset partitions of normal multisets of size n.

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A323788 Number of non-isomorphic weight-n sets of multisets of multisets.

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A323789 Number of non-isomorphic weight-n sets of sets of multisets.

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A323791 Number of non-isomorphic weight-n sets of multisets of sets.

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A323792 Number of non-isomorphic weight-n multisets of sets of sets.

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A323793 Number of non-isomorphic weight-n multisets of multisets of sets.

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