A316983 -id:A316983 - cp's OEIS frontend

A330097 MM-numbers of VDD-normalized multiset partitions.

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[#]]&]

A320797 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 15, 33, 60, 121

Offset: 0

Gus Wiseman, Nov 02 2018

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n and no rows or columns summing to 0 or 1, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 15 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                   {{12}{12}}  {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{11}{23}{23}}  {{22}{11222}}
                                            {{12}{13}{23}}  {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}
Inequivalent representatives of the a(6) = 9 symmetric matrices with no rows or columns summing to 1:
  [6]
.
  [3 0]  [2 1]  [4 0]  [3 1]  [2 2]
  [0 3]  [1 2]  [0 2]  [1 1]  [2 0]
.
  [2 0 0]  [2 0 0]  [1 1 0]
  [0 2 0]  [0 1 1]  [1 0 1]
  [0 0 2]  [0 1 1]  [0 1 1]

Cf. A000219, A007716, A302545, A316980, A316983, A319560, A319616, A319721.

Cf. A320796, A320798, A320799, A320804, A320811, A320812, A320813.

A319779 Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 20, 66, 226, 696, 2156

Offset: 0

Gus Wiseman, Sep 27 2018

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.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			Non-isomorphic representatives of the a(4) = 1 through a(6) = 20 multiset partitions:
4: {{1,3},{2,3}}
5: {{1,2},{2,3,3}}
   {{1,3},{2,3,3}}
   {{1,4},{2,3,4}}
   {{3},{1,3},{2,3}}
6: {{1,2},{2,3,3,3}}
   {{1,3},{2,2,3,3}}
   {{1,3},{2,3,3,3}}
   {{1,3},{2,3,4,4}}
   {{1,4},{2,3,4,4}}
   {{1,5},{2,3,4,5}}
   {{1,1,2},{2,3,3}}
   {{1,2,2},{2,3,3}}
   {{1,2,3},{3,4,4}}
   {{1,2,4},{3,4,4}}
   {{1,2,5},{3,4,5}}
   {{1,3,3},{2,3,3}}
   {{1,3,4},{2,3,4}}
   {{2},{1,2},{2,3,3}}
   {{3},{1,3},{2,3,3}}
   {{4},{1,4},{2,3,4}}
   {{1,3},{2,3},{2,3}}
   {{1,3},{2,3},{3,3}}
   {{1,4},{2,4},{3,4}}
   {{3},{3},{1,3},{2,3}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319775, A319778, A319781, A319783.

A321194 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 17, 12, 3, 1, 40, 35, 12, 3, 1, 125, 112, 45, 12, 3, 1, 354, 347, 148, 45, 12, 3, 1, 1159, 1122, 512, 163, 45, 12, 3, 1, 3774, 3651, 1724, 572, 163, 45, 12, 3, 1, 13113, 12320, 5937, 2020, 593, 163, 45, 12, 3, 1, 46426, 42407, 20492, 7117, 2110, 593, 163, 45, 12, 3, 1

Offset: 1

Gus Wiseman, Oct 29 2018

			Triangle begins:
      1
      3     1
      6     3     1
     17    12     3     1
     40    35    12     3     1
    125   112    45    12     3     1
    354   347   148    45    12     3     1
   1159  1122   512   163    45    12     3     1
   3774  3651  1724   572   163    45    12     3     1
  13113 12320  5937  2020   593   163    45    12     3     1
The fourth row counts the following non-isomorphic multiset partitions.
  {{1,1,1,1}}        {{1,1},{2,2}}      {{1},{2},{3,3}}    {{1},{2},{3},{4}}
  {{1,1,2,2}}        {{1},{2,2,2}}      {{1},{2},{3,4}}
  {{1,2,2,2}}        {{1},{2,3,3}}      {{1},{2},{3},{3}}
  {{1,2,3,3}}        {{1,2},{3,3}}
  {{1,2,3,4}}        {{1},{2,3,4}}
  {{1},{1,1,1}}      {{1,2},{3,4}}
  {{1,1},{1,1}}      {{1},{1},{2,2}}
  {{1},{1,2,2}}      {{1},{1},{2,3}}
  {{1,2},{1,2}}      {{1},{2},{2,2}}
  {{1,2},{2,2}}      {{1},{3},{2,3}}
  {{1,3},{2,3}}      {{1},{1},{2},{2}}
  {{2},{1,2,2}}      {{1},{2},{2},{2}}
  {{3},{1,2,3}}
  {{1},{1},{1,1}}
  {{1},{2},{1,2}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

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

First column is A007718. Row sums are A007716.

Cf. A006126, A048143, A143543, A306005, A316983, A317672, A317674, A319616, A321155.

O.g.f.: Product 1/(1 - t*x^n)^A007718(n).

Terms a(56) and beyond from Andrew Howroyd, Jan 11 2024

A339888 Number of non-isomorphic multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 3, 5, 13, 23, 55, 104, 236, 470, 1039, 2140, 4712, 9962, 21961, 47484, 105464, 232324, 521338, 1167825, 2651453, 6031136, 13863054, 31987058, 74448415, 174109134, 410265423, 971839195, 2317827540, 5558092098, 13412360692, 32542049038, 79424450486

Offset: 0

Gus Wiseman, Jan 09 2021

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 13 multiset partitions:
  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{1}}  {{2},{1,2}}    {{1,2},{3,4}}
         {{1},{2}}  {{1},{1},{1}}  {{1,3},{2,3}}
                    {{1},{2},{2}}  {{1},{1},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{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

The version for set partitions is A000085, with ordered version A080599.
The case of integer partitions is 1 + A004526(n), ranked by A003586.
Non-isomorphic multiset partitions are counted by A007716.
The case without singletons is A007717.
The version allowing non-strict pairs (x,x) is A320663.
A001190 counts rooted trees with out-degrees <= 2, ranked by A292050.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
A339887 counts factorizations into primes or squarefree semiprimes.
Cf. A001055, A007718, A316983, A319616, A320656, A321729, A339740, A339741.

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}
gs(v) = {sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i], v[j])); g*x^(2*v[i]*v[j]/g))) + sum(i=1, #v, my(r=v[i]); (1 + (1+r)%2)*x^r + ((r-1)\2)*x^(2*r))}
a(n)={if(n==0, 1, my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!)} \\ Andrew Howroyd, Apr 16 2021

Terms a(11) and beyond from Andrew Howroyd, Apr 16 2021

A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 7, 8, 9, 6, 3, 12, 13, 14, 15, 16, 3, 18, 19, 12, 21, 6, 7, 24, 9, 26, 27, 28, 13, 30, 3, 32, 15, 6, 35, 36, 37, 38, 39, 24, 3, 42, 13, 12, 45, 14, 13, 48, 49, 18, 15, 52, 53, 54, 15, 56, 57, 26, 3, 60, 37, 6, 63, 64, 39, 30, 3, 12, 35, 70

Offset: 1

Gus Wiseman, Dec 03 2019

nonn

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.
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}}

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330060.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, 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]]}]])]];
Table[Map[Times@@Prime/@#&,sysnorm[primeMS/@primeMS[n]],{0,1}],{n,100}]

A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 2, 14, 814, 1174774, 909125058112, 291200434263385001951232

Offset: 0

Gus Wiseman, Sep 27 2018

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}}.

A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			The a(3) = 14 set systems:
   {{1},{1,2},{1,2,3}}
   {{1},{1,3},{1,2,3}}
   {{2},{1,2},{1,2,3}}
   {{2},{2,3},{1,2,3}}
   {{3},{1,3},{1,2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,2},{1,3},{1,2,3}}
   {{1,2},{2,3},{1,2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{1,2},{1,3},{1,2,3}}
   {{2},{1,2},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A007716, A281116, A283877, A305854, A306006,  A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319769.
Intersecting set-systems are A051185.
The unlabeled multiset partition version is A319773.
The covering case is A327037.
The version without strict dual is A327038.
Cointersecting set-systems are A327039.
The BII-numbers of these set-systems are A327061.
Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[#,Intersection[#1,#2]=={}&]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}] (* Gus Wiseman, Aug 19 2019 *)

a(6)-a(7) from Christian Sievers, Aug 18 2024

A330105 MM-number of the brute-force normalization of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 7, 8, 9, 6, 3, 12, 13, 14, 15, 16, 3, 18, 19, 12, 21, 6, 7, 24, 9, 26, 27, 28, 13, 30, 3, 32, 15, 6, 69, 36, 37, 38, 39, 24, 3, 42, 13, 12, 45, 14, 13, 48, 49, 18, 15, 52, 53, 54, 15, 56, 57, 26, 3, 60, 37, 6, 63, 64, 39, 30, 3, 12, 69, 138

Offset: 1

Gus Wiseman, Dec 02 2019

nonn

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}}.

This sequence is idempotent and its image/fixed points are A330104.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A316983, A317533, A330098, A330103.
Other normalizations: A330061 (VDD MM), A330101 (brute-force BII), A330102 (VDD BII), A330105 (brute-force MM).
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).

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]}];
Table[Map[Times@@Prime/@#&,brute[primeMS/@primeMS[n]],{0,1}],{n,100}]

cp's OEIS Frontend

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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A320797 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.

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A319779 Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.

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A321194 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.

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A339888 Number of non-isomorphic multiset partitions of weight n into singletons or strict pairs.

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A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.

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A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

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