A326248 Number of crossing, nesting set partitions of {1..n}.

0, 0, 0, 0, 0, 2, 28, 252, 1890, 13020, 86564, 571944, 3826230, 26233662, 185746860, 1364083084, 10410773076, 82609104802, 681130756224, 5829231836494, 51711093240518, 474821049202852, 4506533206814480, 44151320870760216, 445956292457725714

Offset

0,6

Comments

A set 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, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t.

Links

Christian Sievers, Table of n, a(n) for n = 0..576

Formula

a(n) = A000110(n) - 2*A000108(n) + A001519(n). - Christian Sievers, Oct 16 2024

Example

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

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
croXQ[stn_]:=MatchQ[stn, {___, {___, x_, y_, ___}, ___, {___, z_, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
nesXQ[stn_]:=MatchQ[stn, {___, {___, x_, y_, ___}, ___, {___, z_, t_, ___}, ___}/; x<z<t<y||z<x<y<t];
Table[Length[Select[sps[Range[n]], nesXQ[#]&&croXQ[#]&]], {n, 0, 8}]

Crossrefs

Crossing and nesting set partitions are (both) A016098.

Crossing, capturing set partitions are A326246.

Nesting, non-crossing set partitions are A122880.

Cf. A000108, A000110, A001519, A058681, A099947, A117662, A324170.

Cf. A326209, A326211, A326243, A326245, A326256, A326258.

Sequence in context: A147537 A183067 A056261 * A230270 A230759 A229581

Adjacent sequences: A326245 A326246 A326247 * A326249 A326250 A326251

Keyword

nonn

Author

Gus Wiseman, Jun 20 2019

Extensions

a(11) and beyond from Christian Sievers, Oct 16 2024

Status

approved