A049311 -id:A049311 - cp's OEIS frontend

A323790 Number of non-isomorphic weight-n sets of sets of sets.

1, 1, 3, 9, 33, 113, 474, 1985

Offset: 0

Gus Wiseman, Jan 27 2019

Non-isomorphic sets of sets are counted by A283877.
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) = 9 sets of sets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{2}}    {{1}{12}}
         {{1}}{{2}}  {{1}{23}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}
Non-isomorphic representatives of the a(4) = 33 sets of sets of sets:
  {{1234}}             {{1}{123}}         {{1}{2}{12}}       {{1}}{{1}{12}}
  {{1}{234}}           {{12}{13}}         {{1}}{{2}{12}}
  {{12}{34}}           {{1}}{{123}}       {{12}}{{1}{2}}
  {{1}}{{234}}         {{1}{2}{13}}       {{1}}{{2}}{{12}}
  {{1}{2}{34}}         {{12}}{{13}}       {{1}}{{2}}{{1}{2}}
  {{12}}{{34}}         {{1}}{{1}{23}}
  {{1}}{{2}{34}}       {{1}}{{2}{13}}
  {{1}{2}{3}{4}}       {{12}}{{1}{3}}
  {{12}}{{3}{4}}       {{2}}{{1}{13}}
  {{1}}{{2}}{{34}}     {{1}}{{1}{2}{3}}
  {{1}}{{2}{3}{4}}     {{1}}{{2}}{{13}}
  {{1}{2}}{{3}{4}}     {{1}{2}}{{1}{3}}
  {{1}}{{2}}{{3}{4}}   {{1}}{{2}}{{1}{3}}
  {{1}}{{2}}{{3}}{{4}}

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.

Cf. A323787, A323788, A323789, A323791, A323792, A323793, A323794, A323795.

A318396 Number of pairs of integer partitions (y, v) of n such that there exists a pair of set partitions of {1,...,n} with meet {{1},...,{n}}, the first having block sizes y and the second v.

Original entry on oeis.org

1, 1, 3, 6, 15, 28, 64, 116, 238, 430, 818, 1426, 2618, 4439, 7775, 12993, 22025, 35946, 59507, 95319, 154073, 243226, 385192, 598531, 933096, 1429794, 2193699, 3322171, 5027995, 7524245, 11253557, 16661211, 24637859, 36130242, 52879638, 76830503, 111422013, 160505622

Offset: 0

Gus Wiseman, Aug 25 2018

nonn

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. a(n) is also the number of combinatory separations (see A269134 for definition) of strongly normal multisets of size n into normal sets.
From Andrew Howroyd, Oct 31 2019: (Start)
Also, the number of distinct unordered row and column sums of binary matrices without empty columns or rows and with a total of n ones. Only matrices in which both row and columns sums are weakly increasing need to be considered.
By the Gale-Ryser theorem this is equivalent to the number of pairs of integer partitions (y,v) of n with y dominating v. (End)

			The a(4) = 15 pairs of integer partitions:
     4, 1111
    22, 22
    22, 211
    22, 1111
    31, 211
    31, 1111
   211, 22
   211, 31
   211, 211
   211, 1111
  1111, 4
  1111, 22
  1111, 31
  1111, 211
  1111, 1111
The a(4) = 15 combinatory separations:
  1111<={1,1,1,1}
  1112<={1,1,12}
  1112<={1,1,1,1}
  1122<={12,12}
  1122<={1,1,12}
  1122<={1,1,1,1}
  1123<={1,123}
  1123<={12,12}
  1123<={1,1,12}
  1123<={1,1,1,1}
  1234<={1234}
  1234<={1,123}
  1234<={12,12}
  1234<={1,1,12}
  1234<={1,1,1,1}

Andrew Howroyd, Table of n, a(n) for n = 0..100
Manfred Krause, A simple proof of the Gale-Ryser theorem, American Mathematical Monthly, 1996.
Wikipedia, Gale-Ryser theorem

Cf. A000041, A000110, A001247, A007716, A008277, A029894, A049311, A059849, A116540, A181939, A265947, A269134, A318393, A318394, A327913.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Select[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}],And@@UnsameQ@@@#[[2]]&]],{n,6}]

IsDom(p,q)=if(#q<#p, 0, my(s=0,t=0); for(i=0, #p-1, s+=p[#p-i]; t+=q[#q-i]; if(t>s, return(0))); 1)
a(n)={if(n<1, n==0, my(s=0); forpart(p=n, forpart(q=n, s+=IsDom(p,q), [1, p[#p]], [#p, n])); s)} \\ Andrew Howroyd, Oct 31 2019

\\ faster version.
a(n)={local(Cache=Map());
  my(recurse(b, c, s, t)=my(hk=Vecsmall([b, c, s, t]), z);
     if(!mapisdefined(Cache, hk, &z),
       z = if(s, sum(i=1, min(s, b), sum(j=1, min(t-s+i, c), self()(i, j, s-i, t-j))),
           if(t, sum(j=1, min(t, c), self()(b, j, s, t-j)), 1));
       mapput(Cache, hk, z)); z);
  recurse(n, n, n, n)
} \\ Andrew Howroyd, Oct 31 2019

Terms a(9) and beyond from Andrew Howroyd, Oct 31 2019

A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 12, 37, 130, 428, 1481, 5091, 17979, 64176, 234311, 869645, 3295100, 12720494, 50083996, 200964437, 821845766, 3423694821, 14524845181, 62725701708, 275629610199, 1231863834775, 5597240308384, 25844969339979, 121224757935416, 577359833539428, 2791096628891679

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(2) = 1 through a(4) = 12 strict multiset partitions with empty intersection:
2: {{1},{2}}
3: {{1},{2,2}}
   {{1},{2,3}}
   {{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},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}

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

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

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

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

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

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

Original entry on oeis.org

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

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.

A321679 Number of non-isomorphic weight-n antichains (not necessarily strict) of sets.

Original entry on oeis.org

1, 1, 3, 5, 12, 19, 45, 75, 170, 314, 713

Offset: 0

Gus Wiseman, Nov 16 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(1) = 1 through a(5) = 19 antichains:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}        {{1,2,3,4,5}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3,4,5}}
         {{1},{2}}  {{1},{1},{1}}  {{1},{2,3,4}}      {{1,2},{3,4,5}}
                    {{1},{2},{2}}  {{1,2},{3,4}}      {{1,4},{2,3,4}}
                    {{1},{2},{3}}  {{1,3},{2,3}}      {{1},{1},{2,3,4}}
                                   {{1},{1},{2,3}}    {{1},{2,3},{2,3}}
                                   {{1},{2},{3,4}}    {{1},{2},{3,4,5}}
                                   {{1},{1},{1},{1}}  {{1},{2,3},{4,5}}
                                   {{1},{1},{2},{2}}  {{1},{2,4},{3,4}}
                                   {{1},{2},{2},{2}}  {{1},{1},{1},{2,3}}
                                   {{1},{2},{3},{3}}  {{1},{2},{2},{3,4}}
                                   {{1},{2},{3},{4}}  {{1},{2},{3},{4,5}}
                                                      {{1},{1},{1},{1},{1}}
                                                      {{1},{1},{2},{2},{2}}
                                                      {{1},{2},{2},{2},{2}}
                                                      {{1},{2},{2},{3},{3}}
                                                      {{1},{2},{3},{3},{3}}
                                                      {{1},{2},{3},{4},{4}}
                                                      {{1},{2},{3},{4},{5}}

Cf. A006126, A049311, A096827, A285572, A293993, A293994, A318099, A319719, A319721, A320799, A321678.

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.

cp's OEIS Frontend

A323790 Number of non-isomorphic weight-n sets of sets of sets.

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A318396 Number of pairs of integer partitions (y, v) of n such that there exists a pair of set partitions of {1,...,n} with meet {{1},...,{n}}, the first having block sizes y and the second v.

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A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

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PARI

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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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PARI

Extensions

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

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Mathematica

PARI

Extensions

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

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A321679 Number of non-isomorphic weight-n antichains (not necessarily strict) of sets.

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

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