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A326260 MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).

%I A326260 #6 Oct 29 2024 19:55:11
%S A326260 2599,4163,5198,6463,6893,7291,7797,8326,8507,9131,9959,10396,10649,
%T A326260 11041,11639,12489,12811,12926,12995,13786,14237,14582,14899,15157,
%U A326260 15594,16123,16403,16652,17014,17063,17089,17141,18101,18193,18262,18643,18659,19337,19389
%N A326260 MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).
%C A326260 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 A326260 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 A326260 The sequence of terms together with their multiset multisystems begins:
%e A326260    2599: {{2,2},{1,2,3}}
%e A326260    4163: {{2,2},{1,2,4}}
%e A326260    5198: {{},{2,2},{1,2,3}}
%e A326260    6463: {{2,2},{1,1,2,3}}
%e A326260    6893: {{1,2,2},{1,2,3}}
%e A326260    7291: {{2,2},{1,2,5}}
%e A326260    7797: {{1},{2,2},{1,2,3}}
%e A326260    8326: {{},{2,2},{1,2,4}}
%e A326260    8507: {{2,3},{1,2,4}}
%e A326260    9131: {{2,2},{1,2,6}}
%e A326260    9959: {{2,2},{1,1,2,4}}
%e A326260   10396: {{},{},{2,2},{1,2,3}}
%e A326260   10649: {{2,2},{1,2,2,3}}
%e A326260   11041: {{1,2,2},{1,2,4}}
%e A326260   11639: {{2,2,2},{1,2,3}}
%e A326260   12489: {{1},{2,2},{1,2,4}}
%e A326260   12811: {{2,2},{1,2,7}}
%e A326260   12926: {{},{2,2},{1,1,2,3}}
%e A326260   12995: {{2},{2,2},{1,2,3}}
%e A326260   13786: {{},{1,2,2},{1,2,3}}
%t A326260 capXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z&&t<y||z<x&&y<t];
%t A326260 nesXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;x<z&&t<y||z<x&&y<t];
%t A326260 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326260 Select[Range[10000],!nesXQ[primeMS/@primeMS[#]]&&capXQ[primeMS/@primeMS[#]]&]
%Y A326260 Non-nesting set partitions are A000108.
%Y A326260 Capturing set partitions are A326243.
%Y A326260 Capturing, non-nesting set partitions are A326249.
%Y A326260 MM-numbers of nesting multiset partitions are A326256.
%Y A326260 MM-numbers of capturing multiset partitions are A326255.
%Y A326260 Cf. A001055, A034827, A056239, A058681, A112798, A117662, A302242, A324170.
%Y A326260 Cf. A326212, A326246, A054391, A326257, A326258, A326259.
%K A326260 nonn
%O A326260 1,1
%A A326260 _Gus Wiseman_, Jun 22 2019