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A326256 MM-numbers of nesting multiset partitions.

%I A326256 #4 Jun 21 2019 22:46:02
%S A326256 667,989,1334,1633,1769,1817,1978,2001,2021,2323,2461,2623,2668,2967,
%T A326256 2987,3197,3266,3335,3538,3634,3713,3749,3956,3979,4002,4042,4171,
%U A326256 4331,4379,4429,4439,4577,4646,4669,4747,4819,4859,4899,4922,4945,5029,5246,5267,5307
%N A326256 MM-numbers of nesting multiset partitions.
%C A326256 First differs from A326255 in lacking 2599.
%C A326256 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 A326256 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 A326256 The sequence of terms together with their multiset multisystems begins:
%e A326256    667: {{2,2},{1,3}}
%e A326256    989: {{2,2},{1,4}}
%e A326256   1334: {{},{2,2},{1,3}}
%e A326256   1633: {{2,2},{1,1,3}}
%e A326256   1769: {{1,3},{1,2,2}}
%e A326256   1817: {{2,2},{1,5}}
%e A326256   1978: {{},{2,2},{1,4}}
%e A326256   2001: {{1},{2,2},{1,3}}
%e A326256   2021: {{1,4},{2,3}}
%e A326256   2323: {{2,2},{1,6}}
%e A326256   2461: {{2,2},{1,1,4}}
%e A326256   2623: {{1,4},{1,2,2}}
%e A326256   2668: {{},{},{2,2},{1,3}}
%e A326256   2967: {{1},{2,2},{1,4}}
%e A326256   2987: {{1,3},{2,2,2}}
%e A326256   3197: {{2,2},{1,7}}
%e A326256   3266: {{},{2,2},{1,1,3}}
%e A326256   3335: {{2},{2,2},{1,3}}
%e A326256   3538: {{},{1,3},{1,2,2}}
%e A326256   3634: {{},{2,2},{1,5}}
%t A326256 nesXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;(x<z&&y>t)||(x>z&&y<t)];
%t A326256 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326256 Select[Range[10000],nesXQ[primeMS/@primeMS[#]]&]
%Y A326256 MM-numbers of crossing multiset partitions are A324170.
%Y A326256 MM-numbers of capturing multiset partitions are A326255.
%Y A326256 Nesting set partitions are A016098.
%Y A326256 Capturing set partitions are A326243.
%Y A326256 Cf. A001055, A034827, A058681, A112798, A117662, A302242.
%Y A326256 Cf. A326211, A326248, A326257, A326258, A326260.
%K A326256 nonn
%O A326256 1,1
%A A326256 _Gus Wiseman_, Jun 20 2019