A317533 -id:A317533 - cp's OEIS frontend

A317755 Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.

0, 1, 6, 30, 130, 629, 2930, 15019, 78224, 438626, 2548481

Offset: 1

Gus Wiseman, Aug 06 2018

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(3) = 6 strongly normal multiset partitions with empty intersection:
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{1},{2}}
  {{1},{2},{3}}

Cf. A007716, A035310, A255906, A317073, A281116, A317077, A317078, A317532, A317533.

Cf. A317748, A317751, A317752, A317757, A317775.

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];
Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]],{n,6}]

a(10)-a(11) from Robert Price, May 08 2021

A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 33, 21, 8, 1, 11, 91, 104, 36, 10, 1, 15, 298, 452, 238, 55, 12, 1, 22, 910, 2335, 1430, 455, 78, 14, 1, 30, 3017, 11992, 10179, 3505, 775, 105, 16, 1, 42, 9945, 66810, 74299, 31881, 7297, 1218, 136, 18, 1, 56

Offset: 1

Gus Wiseman, Jan 27 2019

			Array begins:
      k=1:  k=2:  k=3:  k=4:  k=5:  k=6:
  n=1:  1     1     1     1     1     1
  n=2:  2     4     6     8    10    12
  n=3:  3    10    21    36    55    78
  n=4:  5    33   104   238   455   775
  n=5:  7    91   452  1430  3505  7297
  n=6: 11   298  2335 10179 31881 80897
Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:
  {{111}}          {{112}}          {{123}}
  {{1}{11}}        {{1}{12}}        {{1}{23}}
  {{1}}{{11}}      {{2}{11}}        {{1}}{{23}}
  {{1}{1}{1}}      {{1}}{{12}}      {{1}{2}{3}}
  {{1}}{{1}{1}}    {{1}{1}{2}}      {{1}}{{2}{3}}
  {{1}}{{1}}{{1}}  {{2}}{{11}}      {{1}}{{2}}{{3}}
                   {{1}}{{1}{2}}
                   {{2}}{{1}{1}}
                   {{1}}{{1}}{{2}}

Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).
Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).
Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.

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]]]];
undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Apply[Rule,Table[{undats[m][[i]],i},{i,Length[undats[m]]}],{1}]],First[Sort[expnorm[m,1]]]]];
expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#1>=aft&]}]},Union@@(expnorm[#1,aft+1]&)/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,[__]]],{par,First/@Position[mx,Max[mx]]}]]]];
strnorm[n_]:=(Flatten[MapIndexed[Table[#2,{#1}]&,#1]]&)/@IntegerPartitions[n];
kmp[n_,k_]:=kmp[n,k]=If[k==1,strnorm[n],Union[expnorm/@Join@@mps/@kmp[n,k-1]]];
Table[Length[kmp[sum-k,k]],{sum,1,7},{k,1,sum-1}]

a(46)-a(56) from Robert Price, May 11 2021

A320796 Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 3, 1, 1, 7, 14, 10, 3, 1, 1, 9, 23, 24, 11, 3, 1, 1, 12, 39, 53, 34, 12, 3, 1, 1, 14, 61, 102, 86, 39, 12, 3, 1, 1, 17, 90, 193, 201, 117, 42, 12, 3, 1, 1, 20, 129, 340, 434, 310, 136, 43, 12, 3, 1, 1, 24, 184, 584, 902, 778, 412, 149, 44, 12, 3, 1

Offset: 1

Gus Wiseman, Nov 02 2018

Also the number of nonnegative integer k X k symmetric matrices with sum of elements equal to n and no zero rows or columns, up to row and column permutations.
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}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Triangle begins:
   1
   1   1
   1   2   1
   1   4   3   1
   1   5   7   3   1
   1   7  14  10   3   1
   1   9  23  24  11   3   1
   1  12  39  53  34  12   3   1
   1  14  61 102  86  39  12   3   1
   1  17  90 193 201 117  42  12   3   1
Non-isomorphic representatives of the multiset partitions for n = 1 through 5 (commas elided):
1: {{1}}
.
2: {{11}}  {{1}{2}}
.
3: {{111}}  {{1}{22}}  {{1}{2}{3}}
.           {{2}{12}}
.
4: {{1111}}  {{11}{22}}  {{1}{1}{23}}  {{1}{2}{3}{4}}
.            {{12}{12}}  {{1}{2}{33}}
.            {{1}{222}}  {{1}{3}{23}}
.            {{2}{122}}
.
5: {{11111}}  {{11}{122}}  {{1}{22}{33}}  {{1}{2}{2}{34}}  {{1}{2}{3}{4}{5}}
.             {{11}{222}}  {{1}{23}{23}}  {{1}{2}{3}{44}}
.             {{12}{122}}  {{1}{2}{333}}  {{1}{2}{4}{34}}
.             {{1}{2222}}  {{1}{3}{233}}
.             {{2}{1222}}  {{2}{12}{33}}
.                          {{2}{13}{23}}
.                          {{3}{3}{123}}

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A316983.

Cf. A000219, A007716, A316980, A317533, A318805, A319560, A319616, A319721, A320797-A320813.

row(n)={vector(n, k, T(k,n) - T(k-1,n))} \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

T(n,k) = A318805(k,n) - A318805(k-1,n). - Andrew Howroyd, Jan 16 2024

a(56) onwards from Andrew Howroyd, Jan 16 2024

A330060 MM-numbers of VDD-normalized multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 70, 72, 74, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128

Offset: 1

Gus Wiseman, Dec 03 2019

First differs from A330104 and A330120 in having 35 and lacking 69, with corresponding multisets of multisets 35: {{2},{1,1}} and 69: {{1},{2,2}}.
First differs from A330108 in having 207 and lacking 175, with corresponding multisets of multisets 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all VDD-normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        49: {11}{11}         84: {}{}{1}{11}
   2: {}          24: {}{}{}{1}      52: {}{}{12}         89: {1112}
   3: {1}         26: {}{12}         53: {1111}           90: {}{1}{1}{2}
   4: {}{}        27: {1}{1}{1}      54: {}{1}{1}{1}      91: {11}{12}
   6: {}{1}       28: {}{}{11}       56: {}{}{}{11}       95: {2}{111}
   7: {11}        30: {}{1}{2}       57: {1}{111}         96: {}{}{}{}{}{1}
   8: {}{}{}      32: {}{}{}{}{}     60: {}{}{1}{2}       98: {}{11}{11}
   9: {1}{1}      35: {2}{11}        63: {1}{1}{11}      104: {}{}{}{12}
  12: {}{}{1}     36: {}{}{1}{1}     64: {}{}{}{}{}{}    105: {1}{2}{11}
  13: {12}        37: {112}          70: {}{2}{11}       106: {}{1111}
  14: {}{11}      38: {}{111}        72: {}{}{}{1}{1}    108: {}{}{1}{1}{1}
  15: {1}{2}      39: {1}{12}        74: {}{112}         111: {1}{112}
  16: {}{}{}{}    42: {}{1}{11}      76: {}{}{111}       112: {}{}{}{}{11}
  18: {}{1}{1}    45: {1}{1}{2}      78: {}{1}{12}       113: {123}
  19: {111}       48: {}{}{}{}{1}    81: {1}{1}{1}{1}    114: {}{1}{111}

Equals the image/fixed points of the idempotent sequence A330061.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Select[Range[100],Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]

A330097 MM-numbers of VDD-normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 35, 37, 39, 45, 49, 53, 57, 63, 81, 89, 91, 95, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 245, 247, 259, 265, 267, 273, 281, 285, 311, 315, 329, 333, 339, 343

Offset: 1

Gus Wiseman, Dec 04 2019

nonn

First differs from A330122 in having 207 and lacking 175, with corresponding multiset partitions 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
A multiset partition is a finite multiset of finite nonempty multisets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all VDD-normalized multiset partitions together with their MM-numbers begins:
   1: 0             57: {1}{111}        151: {1122}
   3: {1}           63: {1}{1}{11}      159: {1}{1111}
   7: {11}          81: {1}{1}{1}{1}    161: {11}{22}
   9: {1}{1}        89: {1112}          165: {1}{2}{3}
  13: {12}          91: {11}{12}        169: {12}{12}
  15: {1}{2}        95: {2}{111}        171: {1}{1}{111}
  19: {111}        105: {1}{2}{11}      183: {1}{122}
  21: {1}{11}      111: {1}{112}        189: {1}{1}{1}{11}
  27: {1}{1}{1}    113: {123}           195: {1}{2}{12}
  35: {2}{11}      117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         245: {2}{11}{11}
  53: {1111}       147: {1}{11}{11}     247: {12}{111}
For example, 1155 is the MM-number of {{1},{2},{3},{1,1}}, which is VDD-normalized, so 1155 belongs to the sequence.
On the other hand, 69  is the MM-number of {{1},{2,2}}, but the VDD-normalization is {{2},{1,1}}, so 69 does not belong to the sequence.

Equals the odd terms of A330060.
A subset of A320634.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330061, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]

A330104 MM-numbers of brute-force normalized multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 69, 72, 74, 76, 78, 81, 84, 89, 90, 91, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128, 131, 133

Offset: 1

Gus Wiseman, Dec 02 2019

First differs from A330060 and A330108 in having 69 and lacking 35, with corresponding multisets of multisets 69: {{1},{2,2}} and 35: {{2},{1,1}}.
First differs from A330120 in having 435 and lacking 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all brute-force normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        52: {}{}{12}         89: {1112}
   2: {}          24: {}{}{}{1}      53: {1111}           90: {}{1}{1}{2}
   3: {1}         26: {}{12}         54: {}{1}{1}{1}      91: {11}{12}
   4: {}{}        27: {1}{1}{1}      56: {}{}{}{11}       96: {}{}{}{}{}{1}
   6: {}{1}       28: {}{}{11}       57: {1}{111}         98: {}{11}{11}
   7: {11}        30: {}{1}{2}       60: {}{}{1}{2}      104: {}{}{}{12}
   8: {}{}{}      32: {}{}{}{}{}     63: {1}{1}{11}      105: {1}{2}{11}
   9: {1}{1}      36: {}{}{1}{1}     64: {}{}{}{}{}{}    106: {}{1111}
  12: {}{}{1}     37: {112}          69: {1}{22}         108: {}{}{1}{1}{1}
  13: {12}        38: {}{111}        72: {}{}{}{1}{1}    111: {1}{112}
  14: {}{11}      39: {1}{12}        74: {}{112}         112: {}{}{}{}{11}
  15: {1}{2}      42: {}{1}{11}      76: {}{}{111}       113: {123}
  16: {}{}{}{}    45: {1}{1}{2}      78: {}{1}{12}       114: {}{1}{111}
  18: {}{1}{1}    48: {}{}{}{}{1}    81: {1}{1}{1}{1}    117: {1}{1}{12}
  19: {111}       49: {11}{11}       84: {}{}{1}{11}     120: {}{}{}{1}{2}

Equals the image/fixed points of the idempotent sequence A330105.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A316983, A317533, A330098, A330101, A330103.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[100],Sort[primeMS/@primeMS[#]]==brute[primeMS/@primeMS[#]]&]

A330107 MM-numbers of brute-force normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 37, 39, 45, 49, 53, 57, 63, 69, 81, 89, 91, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 247, 259, 267, 273, 281, 285, 309, 311, 315, 329, 333, 339, 343, 351, 359

Offset: 1

Gus Wiseman, Dec 02 2019

A multiset partition is a finite multiset of finite nonempty multisets of positive integers.
We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all brute-force normalized multiset partitions together with their MM-numbers begins:
   1: 0             63: {1}{1}{11}      159: {1}{1111}
   3: {1}           69: {1}{22}         161: {11}{22}
   7: {11}          81: {1}{1}{1}{1}    165: {1}{2}{3}
   9: {1}{1}        89: {1112}          169: {12}{12}
  13: {12}          91: {11}{12}        171: {1}{1}{111}
  15: {1}{2}       105: {1}{2}{11}      183: {1}{122}
  19: {111}        111: {1}{112}        189: {1}{1}{1}{11}
  21: {1}{11}      113: {123}           195: {1}{2}{12}
  27: {1}{1}{1}    117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         247: {12}{111}
  53: {1111}       147: {1}{11}{11}     259: {11}{112}
  57: {1}{111}     151: {1122}          267: {1}{1112}

Equals the odd terms of A330104.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A300300, A316983, A317533, A320664, A330061, A330098, A330101, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==brute[primeMS/@primeMS[#]]&]

A330108 MM-numbers of MM-normalized multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 70, 72, 74, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128

Offset: 1

Gus Wiseman, Dec 05 2019

First differs from A330060 in having 175 and lacking 207, with corresponding multisets of multisets 175: {{2},{2},{1,1}} and 207: {{1},{1},{2,2}}.
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
We define the MM-normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest MM-number.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}

			The sequence of all MM-normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        49: {11}{11}         84: {}{}{1}{11}
   2: {}          24: {}{}{}{1}      52: {}{}{12}         89: {1112}
   3: {1}         26: {}{12}         53: {1111}           90: {}{1}{1}{2}
   4: {}{}        27: {1}{1}{1}      54: {}{1}{1}{1}      91: {11}{12}
   6: {}{1}       28: {}{}{11}       56: {}{}{}{11}       95: {2}{111}
   7: {11}        30: {}{1}{2}       57: {1}{111}         96: {}{}{}{}{}{1}
   8: {}{}{}      32: {}{}{}{}{}     60: {}{}{1}{2}       98: {}{11}{11}
   9: {1}{1}      35: {2}{11}        63: {1}{1}{11}      104: {}{}{}{12}
  12: {}{}{1}     36: {}{}{1}{1}     64: {}{}{}{}{}{}    105: {1}{2}{11}
  13: {12}        37: {112}          70: {}{2}{11}       106: {}{1111}
  14: {}{11}      38: {}{111}        72: {}{}{}{1}{1}    108: {}{}{1}{1}{1}
  15: {1}{2}      39: {1}{12}        74: {}{112}         111: {1}{112}
  16: {}{}{}{}    42: {}{1}{11}      76: {}{}{111}       112: {}{}{}{}{11}
  18: {}{1}{1}    45: {1}{1}{2}      78: {}{1}{12}       113: {123}
  19: {111}       48: {}{}{}{}{1}    81: {1}{1}{1}{1}    114: {}{1}{111}

Equals the image/fixed points of the idempotent sequence A330194.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[100],Sort[primeMS/@primeMS[#]]==mmnorm[primeMS/@primeMS[#]]&]

A330120 MM-numbers of lexicographically normalized multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 69, 72, 74, 76, 78, 81, 84, 89, 90, 91, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128, 131, 133

Offset: 1

Gus Wiseman, Dec 05 2019

nonn

First differs from A330104 in lacking 435 and having 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the lexicographic normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the lexicographically least of these representatives.
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}

			The sequence of all lexicographically normalized multisets of multisets together with their MM-numbers begins:
   1: 0          21: {1}{11}       52: {}{}{12}        89: {1112}
   2: {}         24: {}{}{}{1}     53: {1111}          90: {}{1}{1}{2}
   3: {1}        26: {}{12}        54: {}{1}{1}{1}     91: {11}{12}
   4: {}{}       27: {1}{1}{1}     56: {}{}{}{11}      96: {}{}{}{}{}{1}
   6: {}{1}      28: {}{}{11}      57: {1}{111}        98: {}{11}{11}
   7: {11}       30: {}{1}{2}      60: {}{}{1}{2}     104: {}{}{}{12}
   8: {}{}{}     32: {}{}{}{}{}    63: {1}{1}{11}     105: {1}{2}{11}
   9: {1}{1}     36: {}{}{1}{1}    64: {}{}{}{}{}{}   106: {}{1111}
  12: {}{}{1}    37: {112}         69: {1}{22}        108: {}{}{1}{1}{1}
  13: {12}       38: {}{111}       72: {}{}{}{1}{1}   111: {1}{112}
  14: {}{11}     39: {1}{12}       74: {}{112}        112: {}{}{}{}{11}
  15: {1}{2}     42: {}{1}{11}     76: {}{}{111}      113: {123}
  16: {}{}{}{}   45: {1}{1}{2}     78: {}{1}{12}      114: {}{1}{111}
  18: {}{1}{1}   48: {}{}{}{}{1}   81: {1}{1}{1}{1}   117: {1}{1}{12}
  19: {111}      49: {11}{11}      84: {}{}{1}{11}    120: {}{}{}{1}{2}

A subset of A320456.
MM-weight is A302242.
Non-isomorphic multiset partitions are A007716.
Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105, A330194.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

A330121 MM-numbers of lexicographically normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 37, 39, 45, 49, 53, 57, 63, 69, 81, 89, 91, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 247, 259, 267, 273, 281, 285, 309, 311, 315, 329, 333, 339, 343, 351, 359

Offset: 1

Gus Wiseman, Dec 05 2019

nonn

First differs from A330107 in lacking 435 and having 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the lexicographic normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the lexicographically least of these representatives.
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}

			The sequence of all lexicographically normalized multiset partitions together with their MM-numbers begins:
   1:               63: {1}{1}{11}      159: {1}{1111}
   3: {1}           69: {1}{22}         161: {11}{22}
   7: {11}          81: {1}{1}{1}{1}    165: {1}{2}{3}
   9: {1}{1}        89: {1112}          169: {12}{12}
  13: {12}          91: {11}{12}        171: {1}{1}{111}
  15: {1}{2}       105: {1}{2}{11}      183: {1}{122}
  19: {111}        111: {1}{112}        189: {1}{1}{1}{11}
  21: {1}{11}      113: {123}           195: {1}{2}{12}
  27: {1}{1}{1}    117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         247: {12}{111}
  53: {1111}       147: {1}{11}{11}     259: {11}{112}
  57: {1}{111}     151: {1122}          267: {1}{1112}

Equals the odd terms of A330120.
A subset of A320634.
MM-weight is A302242.
Non-isomorphic multiset partitions are A007716.
Cf. A056239, A112798, A317533, A320456, A330061, A330098, A330103, A330105, A330194.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

cp's OEIS Frontend

A317755 Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.

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A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

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A320796 Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.

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A330060 MM-numbers of VDD-normalized multisets of multisets.

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A330097 MM-numbers of VDD-normalized multiset partitions.

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A330104 MM-numbers of brute-force normalized multisets of multisets.

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A330107 MM-numbers of brute-force normalized multiset partitions.

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A330108 MM-numbers of MM-normalized multisets of multisets.

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A330120 MM-numbers of lexicographically normalized multisets of multisets.

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