A050343 -id:A050343 - cp's OEIS frontend

A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

1, 1, 1, 3, 4, 7, 12, 19, 30, 49, 77, 119, 186, 286, 438, 670, 1014, 1528, 2300, 3437, 5119, 7603, 11241, 16564, 24343, 35650, 52058, 75820, 110115, 159510, 230522, 332324, 477994, 686044, 982519, 1404243, 2003063, 2851720, 4052429, 5748440, 8140007, 11507125

Offset: 0

Christian G. Bower, Oct 15 1999

nonn

Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.

			4=(4)=(3)+(1)=(3+1)=(2+1)+(1).
From _Gus Wiseman_, Oct 11 2018: (Start)
a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:
  {{1}}  {{2}}  {{3}}      {{4}}        {{5}}        {{6}}
                {{1,2}}    {{1,3}}      {{1,4}}      {{1,5}}
                {{1},{2}}  {{1},{3}}    {{2,3}}      {{2,4}}
                           {{1},{1,2}}  {{1},{4}}    {{1,2,3}}
                                        {{2},{3}}    {{1},{5}}
                                        {{1},{1,3}}  {{2},{4}}
                                        {{2},{1,2}}  {{1},{1,4}}
                                                     {{1},{2,3}}
                                                     {{2},{1,3}}
                                                     {{3},{1,2}}
                                                     {{1},{2},{3}}
                                                     {{1},{2},{1,2}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000
N. J. A. Sloane, Transforms

Cf. A050343-A050350, A089254.
Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.
Row sums of A330462 and of A360764.

g:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)
nn=10;Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k],{k,nn}],{x,0,n}],{n,0,nn}] (* Gus Wiseman, Oct 11 2018 *)

Weigh transform of A000009.

A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 14, 56, 219, 1001, 4588

Offset: 0

Gus Wiseman, Jan 27 2019

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) = 14 multiset 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}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Cf. A002846, A005121, A007716, A050343, A213427, A269134, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.

Cf. A323788, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A323795 Number of non-isomorphic weight-n sets of non-overlapping sets of sets.

Original entry on oeis.org

1, 1, 3, 8, 27, 82, 310, 1163

Offset: 0

Gus Wiseman, Jan 28 2019

Also the number of non-isomorphic set partitions of set-systems 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(4) = 27 multiset partitions:
  {{1}}  {{12}}      {{123}}          {{1234}}
         {{1}{2}}    {{1}{12}}        {{1}{123}}
         {{1}}{{2}}  {{1}{23}}        {{12}{13}}
                     {{1}}{{12}}      {{1}{234}}
                     {{1}}{{23}}      {{12}{34}}
                     {{1}{2}{3}}      {{1}}{{123}}
                     {{1}}{{2}{3}}    {{1}{2}{12}}
                     {{1}}{{2}}{{3}}  {{1}{2}{13}}
                                      {{12}}{{13}}
                                      {{1}}{{234}}
                                      {{1}{2}{34}}
                                      {{12}}{{34}}
                                      {{1}}{{2}{12}}
                                      {{12}}{{1}{2}}
                                      {{1}}{{2}{13}}
                                      {{12}}{{1}{3}}
                                      {{1}}{{2}{34}}
                                      {{1}{2}{3}{4}}
                                      {{12}}{{3}{4}}
                                      {{2}}{{1}{13}}
                                      {{1}}{{2}}{{12}}
                                      {{1}}{{2}}{{13}}
                                      {{1}}{{2}}{{34}}
                                      {{1}}{{2}{3}{4}}
                                      {{1}{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}}{{4}}

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A318566.

Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.

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

Original entry on oeis.org

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.

A330459 Number of set partitions of set-systems with total sum n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 26, 42, 78, 148, 280, 481, 867, 1569, 2742, 4933, 8493, 14857, 25925, 44877, 77022, 132511, 226449, 385396, 657314, 1111115, 1875708, 3157379, 5309439, 8885889, 14861478, 24760339, 41162971, 68328959, 113099231, 186926116, 308230044

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

Number of sets of disjoint nonempty sets of nonempty sets of positive integers with total sum n.

			The a(6) = 26 partitions:
  ((6))  ((15))      ((123))          ((1)(2)(12))
         ((24))      ((1)(14))        ((1))((2)(12))
         ((1)(5))    ((1)(23))        ((12))((1)(2))
         ((2)(4))    ((2)(13))        ((2))((1)(12))
         ((1))((5))  ((3)(12))        ((1))((2))((12))
         ((2))((4))  ((1))((14))
                     ((1))((23))
                     ((1)(2)(3))
                     ((2))((13))
                     ((3))((12))
                     ((1))((2)(3))
                     ((2))((1)(3))
                     ((3))((1)(2))
                     ((1))((2))((3))

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A007713, A050342, A050343, A279375, A279785, A283877, A294617, A330460, A330462, A323787-A323795, A330452-A330459.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And[UnsameQ@@Join@@#,And@@UnsameQ@@@Join@@#]&]],{n,0,10}]

\\ here L is A000009 and BellP is A000110 as series.
L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
seq(n)={my(c=L(n), b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c, k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b,k)*polcoef(r,k)))} \\ Andrew Howroyd, Dec 29 2019

a(n) = Sum_k A330462(n,k) * A000110(k).

Terms a(18) and beyond from Andrew Howroyd, Dec 29 2019

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.

A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 2, 0, 0, 0, 0, 5, 11, 3, 0, 0, 0, 0, 0, 6, 16, 8, 0, 0, 0, 0, 0, 0, 8, 25, 15, 1, 0, 0, 0, 0, 0, 0, 10, 35, 28, 4, 0, 0, 0, 0, 0, 0, 0, 12, 52, 46, 9, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 18 2019

			Triangle begins:
  1
  0  1
  0  1  0
  0  2  1  0
  0  2  2  0  0
  0  3  4  0  0  0
  0  4  6  2  0  0  0
  0  5 11  3  0  0  0  0
  0  6 16  8  0  0  0  0  0
  0  8 25 15  1  0  0  0  0  0
  0 10 35 28  4  0  0  0  0  0  0
  ...
Row n = 7 counts the following set-systems:
  {{7}}      {{1},{6}}      {{1},{2},{4}}
  {{1,6}}    {{2},{5}}      {{1},{2},{1,3}}
  {{2,5}}    {{3},{4}}      {{1},{3},{1,2}}
  {{3,4}}    {{1},{1,5}}
  {{1,2,4}}  {{1},{2,4}}
             {{2},{1,4}}
             {{2},{2,3}}
             {{3},{1,3}}
             {{4},{1,2}}
             {{1},{1,2,3}}
             {{1,2},{1,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

Row sums are A050342.
Column k = 1 is A000009.
Cf. A001970, A050343, A063834, A270995, A271619, A279375, A279785, A283877, A294617, A326031, A330456, A330460, A330463, A360764.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]

L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} (1 + y*x^j)^A000009(j). - Andrew Howroyd, Dec 29 2019

cp's OEIS Frontend

A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

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A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

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A323795 Number of non-isomorphic weight-n sets of non-overlapping sets of sets.

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

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A330459 Number of set partitions of set-systems with total sum 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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A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.

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