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A326255 MM-numbers of capturing multiset partitions.

%I A326255 #9 Oct 29 2024 19:52:33
%S A326255 667,989,1334,1633,1769,1817,1978,2001,2021,2323,2461,2599,2623,2668,
%T A326255 2967,2987,3197,3266,3335,3538,3634,3713,3749,3956,3979,4002,4042,
%U A326255 4163,4171,4331,4379,4429,4439,4577,4646,4669,4747,4819,4859,4899,4922,4945,5029,5198
%N A326255 MM-numbers of capturing multiset partitions.
%C A326255 First differs from A326256 in having 2599.
%C A326255 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 A326255 A multiset partition is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and t < y or z < x and y < t. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.
%e A326255 The sequence of terms together with their multiset multisystems begins:
%e A326255    667: {{2,2},{1,3}}
%e A326255    989: {{2,2},{1,4}}
%e A326255   1334: {{},{2,2},{1,3}}
%e A326255   1633: {{2,2},{1,1,3}}
%e A326255   1769: {{1,3},{1,2,2}}
%e A326255   1817: {{2,2},{1,5}}
%e A326255   1978: {{},{2,2},{1,4}}
%e A326255   2001: {{1},{2,2},{1,3}}
%e A326255   2021: {{1,4},{2,3}}
%e A326255   2323: {{2,2},{1,6}}
%e A326255   2461: {{2,2},{1,1,4}}
%e A326255   2599: {{2,2},{1,2,3}}
%e A326255   2623: {{1,4},{1,2,2}}
%e A326255   2668: {{},{},{2,2},{1,3}}
%e A326255   2967: {{1},{2,2},{1,4}}
%e A326255   2987: {{1,3},{2,2,2}}
%e A326255   3197: {{2,2},{1,7}}
%e A326255   3266: {{},{2,2},{1,1,3}}
%e A326255   3335: {{2},{2,2},{1,3}}
%e A326255   3538: {{},{1,3},{1,2,2}}
%t A326255 capXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z&&y>t||x>z&&y<t];
%t A326255 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326255 Select[Range[10000],capXQ[primeMS/@primeMS[#]]&]
%Y A326255 MM-numbers of crossing multiset partitions are A324170.
%Y A326255 MM-numbers of nesting multiset partitions are A326256.
%Y A326255 MM-numbers of crossing capturing multiset partitions are A326259.
%Y A326255 Capturing set partitions are A326243.
%Y A326255 Cf. A001055, A034827, A058681, A112798, A117662, A122880, A302242.
%Y A326255 Cf. A326211, A326249, A054391, A326257, A326258.
%K A326255 nonn
%O A326255 1,1
%A A326255 _Gus Wiseman_, Jun 20 2019