A330107 -id:A330107 - cp's OEIS frontend

A330122 MM-numbers of MM-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, 175, 183, 189, 195, 223, 225, 243, 245, 247, 259, 265, 267, 273, 281, 285, 311, 315, 329, 333, 339, 343

Offset: 1

Gus Wiseman, Dec 05 2019

nonn

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}}
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 MM-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}      175: {2}{2}{11}
  21: {1}{11}      111: {1}{112}        183: {1}{122}
  27: {1}{1}{1}    113: {123}           189: {1}{1}{1}{11}
  35: {2}{11}      117: {1}{1}{12}      195: {1}{2}{12}
  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}

Equals the odd terms of A330108.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
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).

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[1,100,2],Sort[primeMS/@primeMS[#]]==mmnorm[primeMS/@primeMS[#]]&]

A330123 BII-numbers of MM-normalized set-systems.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 13, 15, 20, 21, 23, 31, 33, 37, 45, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 77, 79, 84, 85, 87, 95, 97, 101, 109, 116, 117, 119, 127, 139, 143, 159, 173, 180, 181, 183, 191, 195, 196, 197, 199, 203, 205, 207, 212, 213, 215, 223, 225, 229

Offset: 1

Gus Wiseman, Dec 05 2019

nonn

First differs from A330110 in lacking 141 and having 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.
A set-system is a finite set of finite nonempty set of positive integers.
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}}
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
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 MM-normalized set-systems together with their BII-numbers begins:
   0: {}                           45: {{1},{3},{1,2},{2,3}}
   1: {{1}}                        52: {{1,2},{1,3},{2,3}}
   3: {{1},{2}}                    53: {{1},{1,2},{1,3},{2,3}}
   4: {{1,2}}                      55: {{1},{2},{1,2},{1,3},{2,3}}
   5: {{1},{1,2}}                  63: {{1},{2},{3},{1,2},{1,3},{2,3}}
   7: {{1},{2},{1,2}}              64: {{1,2,3}}
  11: {{1},{2},{3}}                65: {{1},{1,2,3}}
  13: {{1},{3},{1,2}}              67: {{1},{2},{1,2,3}}
  15: {{1},{2},{3},{1,2}}          68: {{1,2},{1,2,3}}
  20: {{1,2},{1,3}}                69: {{1},{1,2},{1,2,3}}
  21: {{1},{1,2},{1,3}}            71: {{1},{2},{1,2},{1,2,3}}
  23: {{1},{2},{1,2},{1,3}}        75: {{1},{2},{3},{1,2,3}}
  31: {{1},{2},{3},{1,2},{1,3}}    77: {{1},{3},{1,2},{1,2,3}}
  33: {{1},{2,3}}                  79: {{1},{2},{3},{1,2},{1,2,3}}
  37: {{1},{1,2},{2,3}}            84: {{1,2},{1,3},{1,2,3}}

A subset of A326754.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
MM-weight is A302242.
Cf. A000120, A000612, A048793, A056239, A070939, A112798, A300913, A319559, A320456, A326031, A330101, A330102, 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).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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[0,100],Sort[bpe/@bpe[#]]==mmnorm[bpe/@bpe[#]]&]

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}]

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}]

A330101 BII-number of the brute-force normalization of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 7, 1, 3, 3, 11, 33, 19, 19, 15, 4, 5, 33, 19, 20, 21, 37, 23, 5, 7, 19, 15, 37, 23, 51, 31, 4, 33, 5, 19, 20, 37, 21, 23, 5, 19, 7, 15, 37, 51, 23, 31, 20, 37, 37, 51, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69, 69

Offset: 0

Gus Wiseman, Dec 02 2019

nonn

First differs from A330102 at a(148) = 545, A330102(148) = 274, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 274: {{2},{1,3},{1,4}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the brute-force normalization of a set-system 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 sets is first by length and then lexicographically.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

Wikipedia, Idempotence

This sequence is idempotent and its image and fixed points are A330099.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems by vertices are A055621.
Unlabeled set-systems by weight are A283877.
Cf. A000612, A300913, A321405, A330061, A330102, 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).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
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[fbi[fbi/@brute[bpe/@bpe[n]]],{n,0,100}]

A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 7, 1, 3, 3, 11, 33, 19, 19, 15, 4, 5, 33, 19, 20, 21, 37, 23, 5, 7, 19, 15, 37, 23, 51, 31, 4, 33, 5, 19, 20, 37, 21, 23, 5, 19, 7, 15, 37, 51, 23, 31, 20, 37, 37, 51, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69, 69

Offset: 0

Gus Wiseman, Dec 04 2019

nonn

First differs from A330101 at a(148) = 274, A330101(148) = 545, with corresponding set-systems 274: {{2},{1,3},{1,4}} and 545: {{1},{2,3},{2,4}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a set-system 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 sets is first by length and then lexicographically.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			56 is the BII-number of {{3},{1,3},{2,3}}, which has VDD-normalization {{1},{1,2},{1,3}} with BII-number 21, so a(56) = 21.

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330100.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A326754, A330061, A330101.
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).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
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[fbi[fbi/@sysnorm[bpe/@bpe[n]]],{n,0,100}]

A330194 MM-number of the MM-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 05 2019

nonn

First differs from A330105 at a(35) = 35, A330105(35) = 69.
First differs from A330061 at a(175) = 175, A330061(175) = 207.
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}}

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330108.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A056239, A112798, A317533, A320456, A330061, A330098, 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]}];
Table[Map[Times@@Prime/@#&,mmnorm[primeMS/@primeMS[n]],{0,1}],{n,100}]

a(n) <= n.

A330195 BII-number of the BII-normalization of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 7, 1, 3, 3, 11, 12, 13, 13, 15, 4, 5, 12, 13, 20, 21, 22, 23, 5, 7, 13, 15, 22, 23, 30, 31, 4, 12, 5, 13, 20, 22, 21, 23, 5, 13, 7, 15, 22, 30, 23, 31, 20, 22, 22, 30, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69

Offset: 0

Gus Wiseman, Dec 05 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
We define the BII-normalization of a set-system 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 BII-number.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330109.
A subset of A326754.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
BII-weight is A326031.
Cf. A000120, A000612, A007716, A048793, A070939, A319559, A330061, A330101, A330102, 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).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
biinorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],biinorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],fbi[fbi/@#]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Table[fbi[fbi/@biinorm[bpe/@bpe[n]]],{n,0,100}]

a(n) <= n.

cp's OEIS Frontend

A330122 MM-numbers of MM-normalized multiset partitions.

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A330123 BII-numbers of MM-normalized set-systems.

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

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

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A330101 BII-number of the brute-force normalization of the set-system with BII-number n.

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A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

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A330194 MM-number of the MM-normalization of the multiset of multisets with MM-number n.

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A330195 BII-number of the BII-normalization of the set-system with BII-number n.

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