cp's OEIS Frontend

A330105 MM-number of the brute-force normalization of the multiset of multisets with MM-number n.

%I A330105 #5 Dec 03 2019 21:12:53
%S A330105 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 A330105 13,30,3,32,15,6,69,36,37,38,39,24,3,42,13,12,45,14,13,48,49,18,15,52,
%U A330105 53,54,15,56,57,26,3,60,37,6,63,64,39,30,3,12,69,138
%N A330105 MM-number of the brute-force normalization of the multiset of multisets with MM-number n.
%C A330105 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.
%C A330105 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 A330105   Brute-force:    43287: {{1},{2,3},{2,2,4}}
%C A330105   Lexicographic:  43143: {{1},{2,4},{2,2,3}}
%C A330105   VDD:            15515: {{2},{1,3},{1,1,4}}
%C A330105   MM:             15265: {{2},{1,4},{1,1,3}}
%C A330105 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}}.
%t A330105 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]]]];
%t A330105 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330105 Table[Map[Times@@Prime/@#&,brute[primeMS/@primeMS[n]],{0,1}],{n,100}]
%Y A330105 This sequence is idempotent and its image/fixed points are A330104.
%Y A330105 Non-isomorphic multiset partitions are A007716.
%Y A330105 Cf. A000612, A055621, A283877, A316983, A317533, A330098, A330103.
%Y A330105 Other normalizations: A330061 (VDD MM), A330101 (brute-force BII), A330102 (VDD BII), A330105 (brute-force MM).
%Y A330105 Other fixed points:
%Y A330105 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330105 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330105 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330105 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330105 - BII: A330109 (set-systems).
%K A330105 nonn
%O A330105 1,2
%A A330105 _Gus Wiseman_, Dec 02 2019