A318566 -id:A318566 - cp's OEIS frontend

A330453 Number of strict multiset partitions of multiset partitions of integer partitions of n.

1, 1, 3, 9, 23, 62, 161, 410, 1031, 2579, 6359, 15575, 37830, 91241, 218581, 520544, 1232431, 2902644, 6802178, 15866054, 36844016, 85202436, 196251933, 450341874, 1029709478, 2346409350, 5329371142, 12066816905, 27240224766, 61317231288, 137643961196

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

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

			The a(4) = 23 partitions:
  ((4))  ((22))    ((31))      ((211))        ((1111))
         ((2)(2))  ((1)(3))    ((1)(21))      ((1)(111))
                   ((1))((3))  ((2)(11))      ((11)(11))
                               ((1)(1)(2))    ((1))((111))
                               ((1))((21))    ((1)(1)(11))
                               ((2))((11))    ((1))((1)(11))
                               ((1))((1)(2))  ((1)(1)(1)(1))
                               ((2))((1)(1))  ((11))((1)(1))
                                              ((1))((1)(1)(1))

Alois P. Heinz, Table of n, a(n) for n = 0..3853

The not necessarily strict case is A007713.

Cf. A001055, A001970, A050336, A050343, A089259, A261049, A271619, A316980, A318566, A323787-A323795, A330452-A330459, A330461, A330463.

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) a(n):= `if`(n<2, 1, add(a(n-k)*add(b(d)
      *d*(-1)^(k/d+1), d=divisors(k)), k=1..n)/n)
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Jul 18 2021

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],UnsameQ@@#&]],{n,0,10}]

Weigh transform of A001970. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

A323786 Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.

Original entry on oeis.org

1, 0, 2, 3, 19, 39, 200, 615, 2849, 11174, 52377, 239269, 1191090, 6041975, 32275288, 177797719, 1017833092, 6014562272, 36717301665, 230947360981, 1495562098099, 9956230757240, 68070158777759, 477439197541792, 3432259679880648, 25267209686664449

Offset: 0

Gus Wiseman, Jan 28 2019

nonn

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(4) = 19 multiset partitions:
  {{1111}}      {{1112}}      {{1123}}      {{1234}}
  {{11}{11}}    {{1122}}      {{11}{23}}    {{12}{34}}
  {{11}}{{11}}  {{11}{12}}    {{12}{13}}    {{12}}{{34}}
                {{11}{22}}    {{11}}{{23}}
                {{12}{12}}    {{12}}{{13}}
                {{11}}{{12}}
                {{11}}{{22}}
                {{12}}{{12}}
Non-isomorphic representatives of the a(5) = 39 multiset partitions:
  {{11111}}      {{11112}}      {{11123}}      {{11234}}      {{12345}}
  {{11}{111}}    {{11122}}      {{11223}}      {{11}{234}}    {{12}{345}}
  {{11}}{{111}}  {{11}{112}}    {{11}{123}}    {{12}{134}}    {{12}}{{345}}
                 {{11}{122}}    {{11}{223}}    {{23}{114}}
                 {{12}{111}}    {{12}{113}}    {{11}}{{234}}
                 {{12}{112}}    {{12}{123}}    {{12}}{{134}}
                 {{22}{111}}    {{13}{122}}    {{23}}{{114}}
                 {{11}}{{112}}  {{23}{111}}
                 {{11}}{{122}}  {{11}}{{123}}
                 {{12}}{{111}}  {{11}}{{223}}
                 {{12}}{{112}}  {{12}}{{113}}
                 {{22}}{{111}}  {{12}}{{123}}
                                {{13}}{{122}}
                                {{23}}{{111}}

Cf. A007716, A302545, A306186, A317791, A318564, A318566.

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

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp(sExp(A-x*sv(1)))))} \\ Andrew Howroyd, Jan 17 2023

Terms a(8) and beyond from Andrew Howroyd, Jan 17 2023

A330464 Number of non-isomorphic weight-n sets of set-systems with distinct multiset unions.

Original entry on oeis.org

1, 1, 3, 9, 32, 111, 463, 1942

Offset: 0

Gus Wiseman, Dec 26 2019

A set-system is a finite set of finite nonempty sets of positive integers.

As an alternative description, a(n) is the number of non-isomorphic sets of sets of sets with n leaves where the inner sets of sets all have different multiset unions.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 sets:
  {}  {{{1}}}  {{{1,2}}}      {{{1,2,3}}}
               {{{1},{2}}}    {{{1},{1,2}}}
               {{{1}},{{2}}}  {{{1},{2,3}}}
                              {{{1}},{{1,2}}}
                              {{{1}},{{2,3}}}
                              {{{1},{2},{3}}}
                              {{{1}},{{1},{2}}}
                              {{{1}},{{2},{3}}}
                              {{{1}},{{2}},{{3}}}

Non-isomorphic sets of sets are A283877.
Non-isomorphic sets of sets of sets are A323790.
Non-isomorphic set partitions of set-systems are A323795.
Cf. A089259, A141268, A271619, A279375, A279785, A306186, A316980, A317533, A318564, A318565, A318566, A330459, A330472.

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A330453 Number of strict multiset partitions of multiset partitions of integer partitions of n.

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A323786 Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.

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A330464 Number of non-isomorphic weight-n sets of set-systems with distinct multiset unions.

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