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

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

Author

Gus Wiseman, Dec 05 2019

Keywords

• nonn
• eigen

Comments

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}}

Examples

			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}
		

Crossrefs

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).

Programs

Mathematica

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[#]]&]