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%I #13 Feb 19 2019 00:36:06
%S 1,0,0,0,1,0,3,2,14,11,80,85,510
%N Number of self-complementary set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).
%C 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}}. This sequence counts certain self-conjugate set partitions, i.e., fixed points under Callan's conjugation operation.
%H David Callan, <a href="https://arxiv.org/abs/math/0508052">On conjugates for set partitions and integer compositions</a>, arXiv:math/0508052 [math.CO], 2005.
%e The a(6) = 3 through a(9) = 11 self-complementary set partitions with no singletons or cyclical adjacencies:
%e {{135}{246}} {{13}{246}{57}} {{1357}{2468}} {{136}{258}{479}}
%e {{13}{25}{46}} {{15}{246}{37}} {{135}{27}{468}} {{147}{258}{369}}
%e {{14}{25}{36}} {{146}{27}{358}} {{148}{269}{357}}
%e {{147}{258}{36}} {{168}{249}{357}}
%e {{157}{248}{36}} {{13}{258}{46}{79}}
%e {{13}{24}{57}{68}} {{14}{258}{37}{69}}
%e {{13}{25}{47}{68}} {{14}{28}{357}{69}}
%e {{14}{26}{37}{58}} {{16}{258}{37}{49}}
%e {{14}{27}{36}{58}} {{16}{28}{357}{49}}
%e {{15}{26}{37}{48}} {{17}{258}{39}{46}}
%e {{15}{27}{36}{48}} {{18}{29}{357}{46}}
%e {{16}{24}{38}{57}}
%e {{16}{25}{38}{47}}
%e {{17}{28}{35}{46}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
%t Table[Select[sps[Range[n]],And[cmp[#]==Sort[#],Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]
%Y Cf. A000110, A000126, A000296, A001610, A080107, A169985, A261139, A306417 (all self-conjugate set partitions), A324011 (self-complementarity not required), A324013 (adjacencies allowed), A324014 (singletons allowed), A324015.
%K nonn,more
%O 0,7
%A _Gus Wiseman_, Feb 12 2019
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