OFFSET
0,3
COMMENTS
Two blocks of a set partitions double-cross each other if they are of the form {{...a...b...c...},{...x...y...z...}} for some a < x < b < y < c < z or x < a < y < b < z < c.
LINKS
Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016.
EXAMPLE
Most small set partitions are not double-crossing. The smallest that is double-crossing is {{1,3,5},{2,4,6}}.
MATHEMATICA
nonXXQ[stn_]:=!MatchQ[stn, {___, {___, a_, ___, b_, ___, c_, ___}, ___, {___, x_, ___, y_, ___, z_, ___}, ___}/; a<x<b<y<c<z||x<a<y<b<z<c];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[Range[n]], nonXXQ]], {n, 0, 8}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 23 2019
STATUS
approved