A330104 MM-numbers of brute-force 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, 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

1,2

Comments

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}}.

Links

Table of n, a(n) for n=1..64.

Example

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}

Mathematica

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[#]]&]

Crossrefs

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).

Sequence in context: A320456 A330060 A330108 * A330120 A239015 A030706

Adjacent sequences: A330101 A330102 A330103 * A330105 A330106 A330107

Keyword

nonn,eigen

Author

Gus Wiseman, Dec 02 2019

Status

approved