A324011 Number of set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

1, 0, 0, 0, 1, 0, 5, 14, 66, 307, 1554, 8415, 48530, 296582, 1913561, 12988776, 92467629, 688528288, 5349409512, 43270425827, 363680219762, 3170394634443, 28619600156344, 267129951788160, 2574517930001445, 25587989366964056, 261961602231869825

Offset

0,7

Comments

These set partitions are fixed points under Callan's bijection phi on set partitions.

Links

Table of n, a(n) for n=0..26.

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Example

The a(4) = 1, a(6) = 5, and a(7) = 14 set partitions:
  {{13}{24}}  {{135}{246}}    {{13}{246}{57}}
              {{13}{25}{46}}  {{13}{257}{46}}
              {{14}{25}{36}}  {{135}{26}{47}}
              {{14}{26}{35}}  {{135}{27}{46}}
              {{15}{24}{36}}  {{136}{24}{57}}
                              {{136}{25}{47}}
                              {{14}{257}{36}}
                              {{14}{26}{357}}
                              {{146}{25}{37}}
                              {{146}{27}{35}}
                              {{15}{246}{37}}
                              {{15}{247}{36}}
                              {{16}{24}{357}}
                              {{16}{247}{35}}

Mathematica

Table[Select[sps[Range[n]], And[Count[#, {_}]==0, Total[If[First[#]==1&&Last[#]==n, 1, 0]+Count[Subtract@@@Partition[#, 2, 1], -1]&/@#]==0]&]//Length, {n, 0, 10}]

Crossrefs

Cf. A000110, A000126, A000296 (singletons allowed, or adjacencies allowed), A001610, A124323, A169985, A261139, A324012, A324014, A324015.

Sequence in context: A004030 A256413 A346212 * A194994 A166795 A128102

Adjacent sequences: A324008 A324009 A324010 * A324012 A324013 A324014

Keyword

nonn

Author

Gus Wiseman, Feb 12 2019

Extensions

a(11)-a(26) from Alois P. Heinz, Feb 12 2019

Status

approved