A324014 Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

1, 0, 1, 1, 2, 3, 9, 16, 43, 89, 250, 571, 1639

Offset

0,5

Comments

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}.

Links

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

David Callan, On conjugates for set partitions and integer compositions [math.CO].

Example

The  a(3) = 1 through a(6) = 9 self-complementary set partitions with no cyclical adjacencies:
  {{1}{2}{3}}  {{13}{24}}      {{14}{25}{3}}      {{135}{246}}
               {{1}{2}{3}{4}}  {{1}{24}{3}{5}}    {{13}{25}{46}}
                               {{1}{2}{3}{4}{5}}  {{14}{25}{36}}
                                                  {{1}{24}{35}{6}}
                                                  {{13}{2}{46}{5}}
                                                  {{14}{2}{36}{5}}
                                                  {{15}{26}{3}{4}}
                                                  {{1}{25}{3}{4}{6}}
                                                  {{1}{2}{3}{4}{5}{6}}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
Table[Select[sps[Range[n]], And[cmp[#]==Sort[#], Total[If[First[#]==1&&Last[#]==n, 1, 0]+Count[Subtract@@@Partition[#, 2, 1], -1]&/@#]==0]&]//Length, {n, 0, 10}]

Crossrefs

Cf. A000110, A000296, A001610, A080107 (self-complementary), A169985, A324012 (self-conjugate), A324015.

Sequence in context: A220125 A095742 A011951 * A289452 A086771 A156066

Adjacent sequences: A324011 A324012 A324013 * A324015 A324016 A324017

Keyword

nonn,more

Author

Gus Wiseman, Feb 12 2019

Status

approved