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%I #4 Jun 22 2019 23:14:19
%S 8903,15167,16717,17806,18647,20329,20453,21797,22489,25607,26709,
%T 27649,29551,30334,31373,32741,33434,34691,35177,35612,35821,37091,
%U 37133,37294,37969,38243,39493,40658,40906,41449,42011,42949,43594,43817,43873,44515,44861
%N MM-numbers of crossing, capturing 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 multiset partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y. It 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. 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 8903: {{1,3},{2,2,4}}
%e 15167: {{1,3},{2,2,5}}
%e 16717: {{2,4},{1,3,3}}
%e 17806: {{},{1,3},{2,2,4}}
%e 18647: {{1,3},{2,2,6}}
%e 20329: {{1,3},{1,2,2,4}}
%e 20453: {{1,2,3},{1,2,4}}
%e 21797: {{1,1,3},{2,2,4}}
%e 22489: {{1,4},{2,2,5}}
%e 25607: {{1,3},{2,2,7}}
%e 26709: {{1},{1,3},{2,2,4}}
%e 27649: {{1,4},{2,2,6}}
%e 29551: {{1,3},{2,2,8}}
%e 30334: {{},{1,3},{2,2,5}}
%e 31373: {{2,5},{1,3,3}}
%e 32741: {{1,3},{2,2,2,4}}
%e 33434: {{},{2,4},{1,3,3}}
%e 34691: {{1,2,3},{2,2,4}}
%e 35177: {{1,3},{1,2,2,5}}
%e 35612: {{},{},{1,3},{2,2,4}}
%t croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t capXQ[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[100000],capXQ[primeMS/@primeMS[#]]&&croXQ[primeMS/@primeMS[#]]&]
%Y Crossing set partitions are A000108.
%Y Capturing set partitions are A326243.
%Y Crossing, capturing set partitions are A326246.
%Y MM-numbers of crossing multiset partitions are A324170.
%Y MM-numbers of nesting multiset partitions are A326256.
%Y MM-numbers of capturing multiset partitions are A326255.
%Y MM-numbers of unsortable multiset partitions are A326258.
%Y Cf. A001055, A001519, A016098, A056239, A058681, A112798, A122880, A302242.
%Y Cf. A326211, A326245, A326248, A326249, A326254.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 22 2019
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