A324012 - OEIS

A324012

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).

1, 0, 0, 0, 1, 0, 3, 2, 14, 11, 80, 85, 510

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OFFSET

0,7

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}}. This sequence counts certain self-conjugate set partitions, i.e., fixed points under Callan's conjugation operation.

LINKS

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

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

EXAMPLE

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}}

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[#], 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, A001610, A080107, A169985, A261139, A306417 (all self-conjugate set partitions), A324011 (self-complementarity not required), A324013 (adjacencies allowed), A324014 (singletons allowed), A324015.

Sequence in context: A243253 A064536 A361456 * A355259 A231183 A324661

Adjacent sequences: A324009 A324010 A324011 * A324013 A324014 A324015

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Feb 12 2019

STATUS

approved

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