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

Author

Gus Wiseman, Jun 20 2019

Keywords

• nonn

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.

Examples

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

Links

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

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.

Programs

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_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;x
				

Formula

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

Extensions

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