A326259 MM-numbers of crossing, capturing multiset partitions (with empty parts allowed).

8903, 15167, 16717, 17806, 18647, 20329, 20453, 21797, 22489, 25607, 26709, 27649, 29551, 30334, 31373, 32741, 33434, 34691, 35177, 35612, 35821, 37091, 37133, 37294, 37969, 38243, 39493, 40658, 40906, 41449, 42011, 42949, 43594, 43817, 43873, 44515, 44861

Offset: 1

Author

Gus Wiseman, Jun 22 2019

Keywords

• nonn

Comments

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.

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.

Examples

			The sequence of terms together with their multiset multisystems begins:
   8903: {{1,3},{2,2,4}}
  15167: {{1,3},{2,2,5}}
  16717: {{2,4},{1,3,3}}
  17806: {{},{1,3},{2,2,4}}
  18647: {{1,3},{2,2,6}}
  20329: {{1,3},{1,2,2,4}}
  20453: {{1,2,3},{1,2,4}}
  21797: {{1,1,3},{2,2,4}}
  22489: {{1,4},{2,2,5}}
  25607: {{1,3},{2,2,7}}
  26709: {{1},{1,3},{2,2,4}}
  27649: {{1,4},{2,2,6}}
  29551: {{1,3},{2,2,8}}
  30334: {{},{1,3},{2,2,5}}
  31373: {{2,5},{1,3,3}}
  32741: {{1,3},{2,2,2,4}}
  33434: {{},{2,4},{1,3,3}}
  34691: {{1,2,3},{2,2,4}}
  35177: {{1,3},{1,2,2,5}}
  35612: {{},{},{1,3},{2,2,4}}
		

Crossrefs

Crossing set partitions are A000108.

Capturing set partitions are A326243.

Crossing, capturing set partitions are A326246.

MM-numbers of crossing multiset partitions are A324170.

MM-numbers of nesting multiset partitions are A326256.

MM-numbers of capturing multiset partitions are A326255.

MM-numbers of unsortable multiset partitions are A326258.

Cf. A001055, A001519, A016098, A056239, A058681, A112798, A122880, A302242.

Cf. A326211, A326245, A326248, A326249, A054391.

Programs

Mathematica

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
Select[Range[100000],capXQ[primeMS/@primeMS[#]]&&croXQ[primeMS/@primeMS[#]]&]