A326259 - OEIS

%I #6 Oct 29 2024 19:54:51

%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, A054391.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 22 2019