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 capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.
LINKS
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
EXAMPLE
The a(4) = 1 and a(5) = 7 set partitions:
{{1,3},{2,4}} {{1,2,4},{3,5}}
{{1,3},{2,4,5}}
{{1},{2,4},{3,5}}
{{1,3},{2,4},{5}}
{{1,3},{2,5},{4}}
{{1,4},{2},{3,5}}
{{1,4},{2,5},{3}}
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];
capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<t<y||z<x<y<t];
Table[Length[Select[sps[Range[n]], !capXQ[#]&&croXQ[#]&]], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 20 2019
STATUS
approved