A326249 Number of capturing set partitions of {1..n} that are not nesting.

0, 0, 0, 0, 0, 1, 9, 55, 283, 1324, 5838, 24744

Offset

0,7

Comments

Capturing is a weaker condition than nesting. A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. For example, {{1,3,5},{2,4}} is capturing but not nesting, so is counted under a(5).

Links

Table of n, a(n) for n=0..11.

Example

The a(6) = 9 set partitions:
  {{1},{2,4,6},{3,5}}
  {{1,3,5},{2,4},{6}}
  {{1,3,6},{2,4},{5}}
  {{1,3,6},{2,5},{4}}
  {{1,4,6},{2},{3,5}}
  {{1,4,6},{2,5},{3}}
  {{1,3,5},{2,4,6}}
  {{1,2,4,6},{3,5}}
  {{1,3,5,6},{2,4}}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
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];
Table[Length[Select[sps[Range[n]], !nesXQ[#]&&capXQ[#]&]], {n, 0, 8}]

Crossrefs

MM-numbers of capturing, non-nesting multiset partitions are A326260.

Nesting set partitions are A016098.

Capturing set partitions are A326243.

Non-crossing, nesting set partitions are A122880 (conjectured).

Cf. A000110, A054391, A058681, A095661, A099947.

Cf. A326212, A326237, A326245, A326246, A054391, A326255, A326256.

Sequence in context: A068970 A141530 A263478 * A016269 A005770 A030053

Adjacent sequences: A326246 A326247 A326248 * A326250 A326251 A326252

Keyword

nonn,more,changed

Author

Gus Wiseman, Jun 20 2019

Status

approved