A326260 MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).

2599, 4163, 5198, 6463, 6893, 7291, 7797, 8326, 8507, 9131, 9959, 10396, 10649, 11041, 11639, 12489, 12811, 12926, 12995, 13786, 14237, 14582, 14899, 15157, 15594, 16123, 16403, 16652, 17014, 17063, 17089, 17141, 18101, 18193, 18262, 18643, 18659, 19337, 19389

Offset

1,1

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 set partition 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. It is nesting 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.

Links

Table of n, a(n) for n=1..39.

Example

The sequence of terms together with their multiset multisystems begins:
   2599: {{2,2},{1,2,3}}
   4163: {{2,2},{1,2,4}}
   5198: {{},{2,2},{1,2,3}}
   6463: {{2,2},{1,1,2,3}}
   6893: {{1,2,2},{1,2,3}}
   7291: {{2,2},{1,2,5}}
   7797: {{1},{2,2},{1,2,3}}
   8326: {{},{2,2},{1,2,4}}
   8507: {{2,3},{1,2,4}}
   9131: {{2,2},{1,2,6}}
   9959: {{2,2},{1,1,2,4}}
  10396: {{},{},{2,2},{1,2,3}}
  10649: {{2,2},{1,2,2,3}}
  11041: {{1,2,2},{1,2,4}}
  11639: {{2,2,2},{1,2,3}}
  12489: {{1},{2,2},{1,2,4}}
  12811: {{2,2},{1,2,7}}
  12926: {{},{2,2},{1,1,2,3}}
  12995: {{2},{2,2},{1,2,3}}
  13786: {{},{1,2,2},{1,2,3}}

Mathematica

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

Crossrefs

Non-nesting set partitions are A000108.

Capturing set partitions are A326243.

Capturing, non-nesting set partitions are A326249.

MM-numbers of nesting multiset partitions are A326256.

MM-numbers of capturing multiset partitions are A326255.

Cf. A001055, A034827, A056239, A058681, A112798, A117662, A302242, A324170.

Cf. A326212, A326246, A326254, A326257, A326258, A326259.

Sequence in context: A236598 A237163 A203257 * A262799 A183965 A031549

Adjacent sequences: A326257 A326258 A326259 * A326261 A326262 A326263

Keyword

nonn

Author

Gus Wiseman, Jun 22 2019

Status

approved