A326256 MM-numbers of nesting multiset partitions.

667, 989, 1334, 1633, 1769, 1817, 1978, 2001, 2021, 2323, 2461, 2623, 2668, 2967, 2987, 3197, 3266, 3335, 3538, 3634, 3713, 3749, 3956, 3979, 4002, 4042, 4171, 4331, 4379, 4429, 4439, 4577, 4646, 4669, 4747, 4819, 4859, 4899, 4922, 4945, 5029, 5246, 5267, 5307

Offset

1,1

Comments

First differs from A326255 in lacking 2599.

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 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.

Links

Table of n, a(n) for n=1..44.

Example

The sequence of terms together with their multiset multisystems begins:
   667: {{2,2},{1,3}}
   989: {{2,2},{1,4}}
  1334: {{},{2,2},{1,3}}
  1633: {{2,2},{1,1,3}}
  1769: {{1,3},{1,2,2}}
  1817: {{2,2},{1,5}}
  1978: {{},{2,2},{1,4}}
  2001: {{1},{2,2},{1,3}}
  2021: {{1,4},{2,3}}
  2323: {{2,2},{1,6}}
  2461: {{2,2},{1,1,4}}
  2623: {{1,4},{1,2,2}}
  2668: {{},{},{2,2},{1,3}}
  2967: {{1},{2,2},{1,4}}
  2987: {{1,3},{2,2,2}}
  3197: {{2,2},{1,7}}
  3266: {{},{2,2},{1,1,3}}
  3335: {{2},{2,2},{1,3}}
  3538: {{},{1,3},{1,2,2}}
  3634: {{},{2,2},{1,5}}

Mathematica

nesXQ[stn_]:=MatchQ[stn, {___, {___, x_, y_, ___}, ___, {___, z_, t_, ___}, ___}/; (x<z&&y>t)||(x>z&&y<t)];
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], nesXQ[primeMS/@primeMS[#]]&]

Crossrefs

MM-numbers of crossing multiset partitions are A324170.

MM-numbers of capturing multiset partitions are A326255.

Nesting set partitions are A016098.

Capturing set partitions are A326243.

Cf. A001055, A034827, A058681, A112798, A117662, A302242.

Cf. A326211, A326248, A326257, A326258, A326260.

Sequence in context: A172922 A046694 A326255 * A210477 A138563 A327908

Adjacent sequences: A326253 A326254 A326255 * A326257 A326258 A326259

Keyword

nonn

Author

Gus Wiseman, Jun 20 2019

Status

approved