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First differs from A330122 in having 207 and lacking 175, with corresponding multisets ofmultiset multisetspartitions 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.

allocatedMM-numbers of forVDD-normalized Gusmultiset Wisemanpartitions.

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

1,2

First differs from A330122 in having 207 and lacking 175, with corresponding multisets of multisets 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.

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[#]]&]

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).

allocated

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Gus Wiseman, Dec 04 2019

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