A306558 Number of double-crossing set partitions of {1,...,n}.

0, 0, 0, 0, 0, 0, 1, 14, 141, 1267, 10841, 91091, 764092

Offset

0,8

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

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

Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016.

Example

The a(7) = 14 double-crossing set partitions:
  {{1,3,5},{2,4,6,7}}
  {{1,3,6},{2,4,5,7}}
  {{1,4,6},{2,3,5,7}}
  {{1,2,4,6},{3,5,7}}
  {{1,3,4,6},{2,5,7}}
  {{1,3,5,6},{2,4,7}}
  {{1,3,5,7},{2,4,6}}
  {{1},{2,4,6},{3,5,7}}
  {{1,3,5},{2,4,6},{7}}
  {{1,3,5},{2,4,7},{6}}
  {{1,3,6},{2,4,7},{5}}
  {{1,3,6},{2,5,7},{4}}
  {{1,4,6},{2},{3,5,7}}
  {{1,4,6},{2,5,7},{3}}

Mathematica

croXXQ[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]], croXXQ]], {n, 0, 8}]

Crossrefs

Cf. A000108, A000110, A001263, A002061, A005493, A007297, A016098 (crossing set partitions), A054726, A099947, A324166, A324172, A324324.

Sequence in context: A003457 A263822 A016241 * A131583 A011547 A011548

Adjacent sequences: A306555 A306556 A306557 * A306559 A306560 A306561

Keyword

nonn,more

Author

Gus Wiseman, Feb 23 2019

Status

approved