cp's OEIS Frontend

A326243 Number of capturing set partitions of {1..n}.

%I A326243 #15 Oct 29 2024 19:49:26
%S A326243 0,0,0,0,1,11,80,503,2993,17609,105017,644528,4107600,27313805,
%T A326243 189866541,1379728831,10470032837,82833202559,681977545967,
%U A326243 5832430910181,51723181525978,474866750479993,4506706112772881,44151975623559477,445958774322599940,4638590033810841345
%N A326243 Number of capturing set partitions of {1..n}.
%C A326243 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. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.
%H A326243 Eric Marberg, <a href="http://arxiv.org/abs/1203.5738">Crossings and nestings in colored set partitions</a>, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
%F A326243 a(n) = A000110(n) - A054391(n).
%e A326243 The a(5) = 11 capturing set partitions:
%e A326243   {{1,2,5},{3,4}}
%e A326243   {{1,3,4},{2,5}}
%e A326243   {{1,3,5},{2,4}}
%e A326243   {{1,4},{2,3,5}}
%e A326243   {{1,4,5},{2,3}}
%e A326243   {{1,5},{2,3,4}}
%e A326243   {{1},{2,5},{3,4}}
%e A326243   {{1,4},{2,3},{5}}
%e A326243   {{1,5},{2},{3,4}}
%e A326243   {{1,5},{2,3},{4}}
%e A326243   {{1,5},{2,4},{3}}
%t A326243 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A326243 capXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z&&y>t||x>z&&y<t];
%t A326243 Table[Length[Select[sps[Range[n]],capXQ[#]&]],{n,0,8}]
%Y A326243 Non-capturing set partitions are A054391.
%Y A326243 Crossing and nesting set partitions are (both) A016098.
%Y A326243 Cf. A000108, A000110, A001519, A054391, A058681, A122880, A324170.
%Y A326243 Cf. A326209, A326211, A326237, A326246, A326249, A326255, A326259.
%K A326243 nonn
%O A326243 0,6
%A A326243 _Gus Wiseman_, Jun 19 2019
%E A326243 a(12) and beyond from _Christian Sievers_, Aug 23 2024