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%I #4 Jun 21 2019 22:46:02
%S 667,989,1334,1633,1769,1817,1978,2001,2021,2323,2461,2623,2668,2967,
%T 2987,3197,3266,3335,3538,3634,3713,3749,3956,3979,4002,4042,4171,
%U 4331,4379,4429,4439,4577,4646,4669,4747,4819,4859,4899,4922,4945,5029,5246,5267,5307
%N MM-numbers of nesting multiset partitions.
%C First differs from A326255 in lacking 2599.
%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 nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z and t < y or z < x and y < t. This is a stronger condition than capturing, 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 667: {{2,2},{1,3}}
%e 989: {{2,2},{1,4}}
%e 1334: {{},{2,2},{1,3}}
%e 1633: {{2,2},{1,1,3}}
%e 1769: {{1,3},{1,2,2}}
%e 1817: {{2,2},{1,5}}
%e 1978: {{},{2,2},{1,4}}
%e 2001: {{1},{2,2},{1,3}}
%e 2021: {{1,4},{2,3}}
%e 2323: {{2,2},{1,6}}
%e 2461: {{2,2},{1,1,4}}
%e 2623: {{1,4},{1,2,2}}
%e 2668: {{},{},{2,2},{1,3}}
%e 2967: {{1},{2,2},{1,4}}
%e 2987: {{1,3},{2,2,2}}
%e 3197: {{2,2},{1,7}}
%e 3266: {{},{2,2},{1,1,3}}
%e 3335: {{2},{2,2},{1,3}}
%e 3538: {{},{1,3},{1,2,2}}
%e 3634: {{},{2,2},{1,5}}
%t nesXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;(x<z&&y>t)||(x>z&&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[#]]&]
%Y MM-numbers of crossing multiset partitions are A324170.
%Y MM-numbers of capturing multiset partitions are A326255.
%Y Nesting set partitions are A016098.
%Y Capturing set partitions are A326243.
%Y Cf. A001055, A034827, A058681, A112798, A117662, A302242.
%Y Cf. A326211, A326248, A326257, A326258, A326260.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 20 2019
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