A326260 - OEIS

%I #6 Oct 29 2024 19:55:11

%S 2599,4163,5198,6463,6893,7291,7797,8326,8507,9131,9959,10396,10649,

%T 11041,11639,12489,12811,12926,12995,13786,14237,14582,14899,15157,

%U 15594,16123,16403,16652,17014,17063,17089,17141,18101,18193,18262,18643,18659,19337,19389

%N MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).

%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 multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

%C A set partition is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and y > t or x > z and y < t. It is nesting if it has two blocks of the form {...x,y...} and {...z,t...} where x < z and y > t or x > z and y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

%e The sequence of terms together with their multiset multisystems begins:

%e    2599: {{2,2},{1,2,3}}

%e    4163: {{2,2},{1,2,4}}

%e    5198: {{},{2,2},{1,2,3}}

%e    6463: {{2,2},{1,1,2,3}}

%e    6893: {{1,2,2},{1,2,3}}

%e    7291: {{2,2},{1,2,5}}

%e    7797: {{1},{2,2},{1,2,3}}

%e    8326: {{},{2,2},{1,2,4}}

%e    8507: {{2,3},{1,2,4}}

%e    9131: {{2,2},{1,2,6}}

%e    9959: {{2,2},{1,1,2,4}}

%e   10396: {{},{},{2,2},{1,2,3}}

%e   10649: {{2,2},{1,2,2,3}}

%e   11041: {{1,2,2},{1,2,4}}

%e   11639: {{2,2,2},{1,2,3}}

%e   12489: {{1},{2,2},{1,2,4}}

%e   12811: {{2,2},{1,2,7}}

%e   12926: {{},{2,2},{1,1,2,3}}

%e   12995: {{2},{2,2},{1,2,3}}

%e   13786: {{},{1,2,2},{1,2,3}}

%t capXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z&&t<y||z<x&&y<t];

%t nesXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;x<z&&t<y||z<x&&y<t];

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

%t Select[Range[10000],!nesXQ[primeMS/@primeMS[#]]&&capXQ[primeMS/@primeMS[#]]&]

%Y Non-nesting set partitions are A000108.

%Y Capturing set partitions are A326243.

%Y Capturing, non-nesting set partitions are A326249.

%Y MM-numbers of nesting multiset partitions are A326256.

%Y MM-numbers of capturing multiset partitions are A326255.

%Y Cf. A001055, A034827, A056239, A058681, A112798, A117662, A302242, A324170.

%Y Cf. A326212, A326246, A054391, A326257, A326258, A326259.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 22 2019