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A324012
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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).
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7
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1, 0, 0, 0, 1, 0, 3, 2, 14, 11, 80, 85, 510
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OFFSET
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0,7
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The a(6) = 3 through a(9) = 11 self-complementary set partitions with no singletons or cyclical adjacencies:
{{135}{246}} {{13}{246}{57}} {{1357}{2468}} {{136}{258}{479}}
{{13}{25}{46}} {{15}{246}{37}} {{135}{27}{468}} {{147}{258}{369}}
{{14}{25}{36}} {{146}{27}{358}} {{148}{269}{357}}
{{147}{258}{36}} {{168}{249}{357}}
{{157}{248}{36}} {{13}{258}{46}{79}}
{{13}{24}{57}{68}} {{14}{258}{37}{69}}
{{13}{25}{47}{68}} {{14}{28}{357}{69}}
{{14}{26}{37}{58}} {{16}{258}{37}{49}}
{{14}{27}{36}{58}} {{16}{28}{357}{49}}
{{15}{26}{37}{48}} {{17}{258}{39}{46}}
{{15}{27}{36}{48}} {{18}{29}{357}{46}}
{{16}{24}{38}{57}}
{{16}{25}{38}{47}}
{{17}{28}{35}{46}}
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MATHEMATICA
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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[#], Count[#, {_}]==0, Total[If[First[#]==1&&Last[#]==n, 1, 0]+Count[Subtract@@@Partition[#, 2, 1], -1]&/@#]==0]&]//Length, {n, 0, 10}]
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CROSSREFS
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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.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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