A318566 -id:A318566 - cp's OEIS frontend

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

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.

A323794 Number of non-isomorphic weight-n multisets of sets of multisets.

Original entry on oeis.org

1, 1, 5, 17, 77, 318, 1561, 7667

Offset: 0

Gus Wiseman, Jan 28 2019

Also the number of non-isomorphic set multipartitions 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) = 17 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{1}}  {{1}{11}}
         {{1}}{{2}}  {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{1}}{{1}}
                     {{1}}{{1}}{{2}}
                     {{1}}{{2}}{{3}}

Cf. A007716, A049311, A283877, A306186, A316980, A317791, A318565, A318566.

Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323795.

A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 3, 8, 7, 7, 8, 11, 12, 15, 5, 15, 17, 22, 14, 27, 19, 30, 13, 27, 30, 33, 26, 42, 37, 56, 7, 44, 45, 51, 34, 77, 67, 72, 25

Offset: 1

Gus Wiseman, Aug 28 2018

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.

The prime indices of n are the n-th row of A296150.

			The a(18) = 17 combinatory separations of {1,1,2,2,3}:
  {11223}
  {1,1122} {1,1123} {1,1223} {11,112} {12,112} {12,122} {12,123}
  {1,1,112} {1,1,122} {1,1,123} {1,11,11} {1,11,12} {1,12,12}
  {1,1,1,11} {1,1,1,12}
  {1,1,1,1,1}

Cf. A007716, A056239, A255906, A269134, A296150, A305936.

Cf. A317791, A318283, A318284, A318285, A318559, A318562, A318566, A318567.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union[Sort/@Map[normize,mps[nrmptn[n]],{2}]]],{n,20}]

A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

Original entry on oeis.org

1, 4, 10, 32, 80, 239, 605, 1670, 4251

Offset: 1

Gus Wiseman, Aug 29 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

			The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318563, A318564, A318565, A318566.

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]]}];
strnormQ[m_]:=OrderedQ[Length/@Split[m],GreaterEqual];
Table[Length[Select[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}],And@@strnormQ/@#[[2]]&]],{n,6}]

A318563 Number of combinatory separations of strongly normal multisets of weight n.

Original entry on oeis.org

1, 4, 10, 33, 85, 272, 730, 2197, 6133

Offset: 1

Gus Wiseman, Aug 29 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

			The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318562, A318564, A318565, A318566.

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[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}]],{n,7}]

A324930 Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 2, 2, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 0, 2, 1, 3, 2, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 3, 2, 1

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The finite multiset of finite multisets of finite multisets of positive integers with MMM number n is obtained by factoring n into prime numbers, then factoring each of their prime indices into prime numbers, then factoring each of their prime indices into prime numbers, and finally taking their prime indices.

			The sequence of all finite multisets of finite multisets of finite multisets of positive integers begins (o is the empty multiset):
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((1)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((1)))
  11: (((2)))
  12: (oo(o))
  13: ((o(1)))
  14: (o(oo))
  15: ((o)((1)))
  16: (oooo)
  17: (((11)))
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((1)))

Cf. A000081, A000720, A001222, A050338, A056239, A112798, A301595, A302242, A318564, A318565, A318566, A324928.

fi[n_]:=If[n==1,{},FactorInteger[n]];
Table[Total[Cases[fi[n],{p_,k_}:>k*Total[Cases[fi[PrimePi[p]],{q_,j_}:>j*PrimeOmega[PrimePi[q]]]]]],{n,60}]

A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

Original entry on oeis.org

1, 0, 1, 0, 4, 2, 0, 10, 8, 3, 0, 33, 48, 18, 5, 0, 91, 204, 118, 32, 7, 0, 298, 959, 743, 266, 58, 11, 0, 910, 4193, 4334, 1927, 519, 94, 15, 0, 3017, 18947, 25305, 13992, 4407, 966, 154, 22, 0, 9945, 84798, 145033, 97947, 36410, 9023, 1679, 236, 30

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
   1
   0   1
   0   4   2
   0  10   8   3
   0  33  48  18   5
   0  91 204 118  32   7
   0 298 959 743 266  58  11
For example, row n = 3 counts the following multiset partitions:
  {{111}}      {{1}}{{11}}    {{1}}{{1}}{{1}}
  {{112}}      {{1}}{{12}}    {{1}}{{1}}{{2}}
  {{123}}      {{1}}{{23}}    {{1}}{{2}}{{3}}
  {{1}{11}}    {{2}}{{11}}
  {{1}{12}}    {{1}}{{1}{1}}
  {{1}{23}}    {{1}}{{1}{2}}
  {{2}{11}}    {{1}}{{2}{3}}
  {{1}{1}{1}}  {{2}}{{1}{1}}
  {{1}{1}{2}}
  {{1}{2}{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..350

Row sums are A318566.
Column k = 1 is A007716 (for n > 0).
Column k = n is A000041.
Partitions of partitions of partitions are A007713.
Twice-factorizations are A050336.
If this is the 3-dimensional version, the 2-dimensional version is A317533.
See A330473 for a variation.
Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

\\ See links in A339645 for combinatorial species functions.
ColGf(k,n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(A,k,x)*x^k + O(x*x^n), sExp(A)) ))}
M(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 17 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 17 2023

A318816 Regular tetrangle where T(n,k,i) is the number of non-isomorphic multiset partitions of length i of multiset partitions of length k of multisets of size n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 3, 4, 3, 5, 14, 14, 9, 20, 9, 5, 14, 9, 5, 7, 28, 28, 33, 80, 33, 16, 68, 52, 16, 7, 28, 33, 16, 7, 11, 69, 69, 104, 266, 104, 74, 356, 282, 74, 29, 199, 253, 118, 29, 11, 69, 104, 74, 29, 11, 15, 134, 134, 294, 800, 294, 263, 1427, 1164

Offset: 1

Gus Wiseman, Sep 04 2018

			Tetrangle begins:
  1   2     3        5             7
      2 2   4 4     14 14         28 28
            3 4 3    9 20  9      33 80 33
                     5 14  9  5   16 68 52 16
                                   7 28 33 16  7
Non-isomorphic representatives of the T(4,3,2) = 20 multiset partitions:
  {{{1}},{{1},{1,1}}}  {{{1,1}},{{1},{1}}}
  {{{1}},{{1},{1,2}}}  {{{1,1}},{{1},{2}}}
  {{{1}},{{1},{2,2}}}  {{{1,1}},{{2},{2}}}
  {{{1}},{{1},{2,3}}}  {{{1,1}},{{2},{3}}}
  {{{1}},{{2},{1,1}}}  {{{1,2}},{{1},{1}}}
  {{{1}},{{2},{1,2}}}  {{{1,2}},{{1},{2}}}
  {{{1}},{{2},{1,3}}}  {{{1,2}},{{1},{3}}}
  {{{1}},{{2},{3,4}}}  {{{1,2}},{{3},{4}}}
  {{{2}},{{1},{1,1}}}  {{{2,3}},{{1},{1}}}
  {{{2}},{{1},{1,3}}}
  {{{2}},{{3},{1,1}}}

Cf. A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318393, A318399, A318564, A318565, A318566.

A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 3, 8, 10, 0, 5, 28, 38, 33, 0, 7, 56, 146, 152, 91, 0, 11, 138, 474, 786, 628, 298, 0, 15, 268, 1388, 3117, 3808, 2486, 910, 0, 22, 570, 3843, 11830, 19147, 18395, 9986, 3017, 0, 30, 1072, 10094, 40438, 87081, 110164, 86388, 39889, 9945

Offset: 0

Gus Wiseman, Dec 20 2019

As an alternative description, T(n,k) is the number of non-isomorphic multisets of nonempty multisets of nonempty multisets with n leaves whose multiset union consists of k multisets.

			Triangle begins:
   1
   0   1
   0   2   4
   0   3   8  10
   0   5  28  38  33
   0   7  56 146 152  91
   0  11 138 474 786 628 298
For example, row n = 3 counts the following multiset partitions:
  {{111}}  {{1}{11}}    {{1}{1}{1}}
  {{112}}  {{1}{12}}    {{1}{1}{2}}
  {{123}}  {{1}{23}}    {{1}{2}{3}}
           {{2}{11}}    {{1}}{{1}{1}}
           {{1}}{{11}}  {{1}}{{1}{2}}
           {{1}}{{12}}  {{1}}{{2}{3}}
           {{1}}{{23}}  {{2}}{{1}{1}}
           {{2}}{{11}}  {{1}}{{1}}{{1}}
                        {{1}}{{1}}{{2}}
                        {{1}}{{2}}{{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..350

Row sums are A318566.
Column k = 1 is A000041 (for n > 0).
Column k = n is A007716.
Partitions of partitions of partitions are A007713.
Twice-factorizations are A050336.
The 2-dimensional version is A317533.
See A330472 for a variation.
Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

\\ See links in A339645 for combinatorial species functions.
ColGf(k, n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(sExp(A), k, x)*x^k + O(x*x^n), A) ))}
M(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 18 2023

Terms a(36) and beyond from Andrew Howroyd, Jan 18 2023

cp's OEIS Frontend

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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A323794 Number of non-isomorphic weight-n multisets of sets of multisets.

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A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

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A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

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Mathematica

A318563 Number of combinatory separations of strongly normal multisets of weight n.

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Mathematica

A324930 Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

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Mathematica

A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

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A318816 Regular tetrangle where T(n,k,i) is the number of non-isomorphic multiset partitions of length i of multiset partitions of length k of multisets of size n.

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A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

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