cp's OEIS Frontend

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

%I A330061 #6 Dec 05 2019 08:23:21
%S A330061 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,
%T A330061 13,30,3,32,15,6,35,36,37,38,39,24,3,42,13,12,45,14,13,48,49,18,15,52,
%U A330061 53,54,15,56,57,26,3,60,37,6,63,64,39,30,3,12,35,70
%N A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.
%C A330061 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.
%C A330061 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}}.
%C A330061 For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
%C A330061   Brute-force:    43287: {{1},{2,3},{2,2,4}}
%C A330061   Lexicographic:  43143: {{1},{2,4},{2,2,3}}
%C A330061   VDD:            15515: {{2},{1,3},{1,1,4}}
%C A330061   MM:             15265: {{2},{1,4},{1,1,3}}
%H A330061 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent">Idempotence</a>
%t A330061 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330061 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]]]];
%t A330061 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]]}]])]];
%t A330061 Table[Map[Times@@Prime/@#&,sysnorm[primeMS/@primeMS[n]],{0,1}],{n,100}]
%Y A330061 This sequence is idempotent and its image/fixed points are A330060.
%Y A330061 Non-isomorphic multiset partitions are A007716.
%Y A330061 MM-weight is A302242.
%Y A330061 Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330098, A330102, A330103, A330105.
%Y A330061 Other fixed points:
%Y A330061 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330061 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330061 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330061 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330061 - BII: A330109 (set-systems).
%K A330061 nonn
%O A330061 1,2
%A A330061 _Gus Wiseman_, Dec 03 2019