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A324013
Number of self-complementary set partitions of {1, ..., n} with no singletons.
3
1, 0, 1, 1, 4, 3, 15, 16, 75, 89, 428, 571, 2781, 4060, 20093, 31697, 159340, 268791, 1372163, 2455804, 12725447, 24012697, 126238060, 249880687, 1332071241, 2754348360, 14881206473, 32029000641, 175297058228, 391548016475, 2169832010759
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
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
FORMULA
From Andrew Howroyd, Feb 16 2022: (Start)
a(2*n) = A086365(n-1) for n > 0.
a(2*n) = n!*[x^n] exp((exp(2*x) - 3)/2 - x + exp(x));
a(2*n+1) = n!*[x^n] (exp(x) - 1)*exp((exp(2*x) - 3)/2 - x + exp(x)).
(End)
EXAMPLE
The a(3) = 1 through a(6) = 15 self-complementary set partitions with no singletons:
{{123}} {{1234}} {{12345}} {{123456}}
{{12}{34}} {{135}{24}} {{123}{456}}
{{13}{24}} {{15}{234}} {{124}{356}}
{{14}{23}} {{1256}{34}}
{{1346}{25}}
{{135}{246}}
{{145}{236}}
{{16}{2345}}
{{12}{34}{56}}
{{13}{25}{46}}
{{14}{25}{36}}
{{15}{26}{34}}
{{16}{23}{45}}
{{16}{24}{35}}
{{16}{25}{34}}
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]&]//Length, {n, 0, 10}]
PROG
(PARI) seq(n)={my(x=x+O(x*x^(n\2)), p=exp((exp(2*x)-3)/2-x+exp(x)), q=(exp(x)-1)*p); vector(n+1, n, my(c=(n-1)\2); c!*polcoef(if(n%2, p, q), c))} \\ Andrew Howroyd, Feb 16 2022
CROSSREFS
Cf. A000110, A000296, A080107 (self-complementary), A086365, A124323, A324012 (self-conjugate).
Sequence in context: A095332 A195586 A120461 * A345013 A113204 A270188
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 12 2019
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Feb 16 2022
STATUS
approved