OFFSET
0,5
COMMENTS
A set partition is totally crossing if every pair of distinct blocks is of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y.
EXAMPLE
The a(6) = 18 totally crossing set partitions:
{{1,2,3,4,5,6}}
{{1,4,6},{2,3,5}}
{{1,4,5},{2,3,6}}
{{1,3,6},{2,4,5}}
{{1,3,5},{2,4,6}}
{{1,3,4},{2,5,6}}
{{1,2,5},{3,4,6}}
{{1,2,4},{3,5,6}}
{{4,6},{1,2,3,5}}
{{3,6},{1,2,4,5}}
{{3,5},{1,2,4,6}}
{{2,6},{1,3,4,5}}
{{2,5},{1,3,4,6}}
{{2,4},{1,3,5,6}}
{{1,5},{2,3,4,6}}
{{1,4},{2,3,5,6}}
{{1,3},{2,4,5,6}}
{{1,4},{2,5},{3,6}}
MATHEMATICA
nn=6;
nonXQ[stn_]:=!MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
stableSets[u_, Q_]:=If[Length[u]===0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r===w||Q[r, w]||Q[w, r]], Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], Intersection[#1, #2]!={}||nonXQ[{#1, #2}]&], Union@@#==Range[n]&]], {n, 0, nn}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 17 2019
STATUS
approved