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A330060
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MM-numbers of VDD-normalized multisets of multisets.
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19
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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
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OFFSET
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1,2
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COMMENTS
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First differs from A330104 and A330120 in having 35 and lacking 69, with corresponding multisets of multisets 35: {{2},{1,1}} and 69: {{1},{2,2}}.
First differs from A330108 in having 207 and lacking 175, with corresponding multisets of multisets 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
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 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}}.
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LINKS
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EXAMPLE
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The sequence of all VDD-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}
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MATHEMATICA
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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[100], Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]
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CROSSREFS
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Equals the image/fixed points of the idempotent sequence A330061.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A330098, A330102, A330103, A330105.
Other fixed points:
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KEYWORD
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nonn,eigen
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AUTHOR
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STATUS
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approved
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