A324166 Number of totally crossing set partitions of {1,...,n}.

1, 1, 1, 1, 2, 6, 18, 57, 207, 842, 3673, 17062, 84897

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.

Links

Table of n, a(n) for n=0..12.

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

Cf. A000108 (non-crossing partitions), A000110, A000296, A002662, A016098 (crossing partitions), A054726, A099947 (topologically connected partitions), A305854, A306006, A306418, A306438, A319752.

Cf. A324170, A324171, A324172, A324173.

Sequence in context: A273203 A148458 A148459 * A355388 A351787 A307755

Adjacent sequences: A324163 A324164 A324165 * A324167 A324168 A324169

Keyword

nonn,more

Author

Gus Wiseman, Feb 17 2019

Status

approved