A326254 Number of non-capturing set partitions of {1..n}.

1, 1, 2, 5, 14, 41, 123, 374, 1147, 3538, 10958, 34042, 105997

Offset

0,3

Comments

Conjectured to be equal to A054391.

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. 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=0..12.

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

Formula

a(n) = A000110(n) - A326243(n).

Example

The a(0) = 1 through a(4) = 14 non-capturing set partitions:
  {}  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
             {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
                        {{1,2},{3}}    {{1,2},{3,4}}
                        {{1,3},{2}}    {{1,2,3},{4}}
                        {{1},{2},{3}}  {{1,2,4},{3}}
                                       {{1,3},{2,4}}
                                       {{1,3,4},{2}}
                                       {{1},{2},{3,4}}
                                       {{1},{2,3},{4}}
                                       {{1,2},{3},{4}}
                                       {{1},{2,4},{3}}
                                       {{1,3},{2},{4}}
                                       {{1,4},{2},{3}}
                                       {{1},{2},{3},{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&&y>t||x>z&&y<t];
Table[Length[Select[sps[Range[n]], !capXQ[#]&]], {n, 0, 5}]

Crossrefs

Capturing set partitions are A326243.

Non-crossing set partitions are A000108.

Cf. A000110, A001519, A016098, A054391, A058681, A099947, A122880.

Cf. A326212, A326237, A326245, A326246, A326249, A326255, A326256, A326260.

Sequence in context: A088355 A113485 A360569 * A054391 A365508 A176677

Adjacent sequences: A326251 A326252 A326253 * A326255 A326256 A326257

Keyword

nonn,more

Author

Gus Wiseman, Jun 20 2019

Status

approved