A324013 Number of self-complementary set partitions of {1, ..., n} with no singletons.

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

Andrew Howroyd, Table of n, a(n) for n = 0..500

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

Adjacent sequences: A324010 A324011 A324012 * A324014 A324015 A324016

Keyword

nonn

Author

Gus Wiseman, Feb 12 2019

Extensions

Terms a(13) and beyond from Andrew Howroyd, Feb 16 2022

Status

approved