A330122 - OEIS

%I #6 Dec 05 2019 17:41:40

%S 1,3,7,9,13,15,19,21,27,35,37,39,45,49,53,57,63,81,89,91,95,105,111,

%T 113,117,131,133,135,141,147,151,159,161,165,169,171,175,183,189,195,

%U 223,225,243,245,247,259,265,267,273,281,285,311,315,329,333,339,343

%N MM-numbers of MM-normalized multiset partitions.

%C We define the MM-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 representative with the smallest MM-number.

%C 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   Brute-force:    43287: {{1},{2,3},{2,2,4}}

%C   Lexicographic:  43143: {{1},{2,4},{2,2,3}}

%C   VDD:            15515: {{2},{1,3},{1,1,4}}

%C   MM:             15265: {{2},{1,4},{1,1,3}}

%C 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}}.

%e The sequence of all MM-normalized multiset partitions together with their MM-numbers begins:

%e    1: 0             57: {1}{111}        151: {1122}

%e    3: {1}           63: {1}{1}{11}      159: {1}{1111}

%e    7: {11}          81: {1}{1}{1}{1}    161: {11}{22}

%e    9: {1}{1}        89: {1112}          165: {1}{2}{3}

%e   13: {12}          91: {11}{12}        169: {12}{12}

%e   15: {1}{2}        95: {2}{111}        171: {1}{1}{111}

%e   19: {111}        105: {1}{2}{11}      175: {2}{2}{11}

%e   21: {1}{11}      111: {1}{112}        183: {1}{122}

%e   27: {1}{1}{1}    113: {123}           189: {1}{1}{1}{11}

%e   35: {2}{11}      117: {1}{1}{12}      195: {1}{2}{12}

%e   37: {112}        131: {11111}         223: {11112}

%e   39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}

%e   45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}

%e   49: {11}{11}     141: {1}{23}         245: {2}{11}{11}

%e   53: {1111}       147: {1}{11}{11}     247: {12}{111}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];

%t brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];

%t Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==mmnorm[primeMS/@primeMS[#]]&]

%Y Equals the odd terms of A330108.

%Y A subset of A320456.

%Y Non-isomorphic multiset partitions are A007716.

%Y MM-weight is A302242.

%Y Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105, A330194.

%Y Other fixed points:

%Y - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).

%Y - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).

%Y - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).

%Y - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).

%Y - BII: A330109 (set-systems).

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 05 2019