A086365 -id:A086365 - cp's OEIS frontend

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

Gus Wiseman, Feb 12 2019

nonn

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}}.

			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}}

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.

Cf. A000110, A000296, A080107 (self-complementary), A086365, A124323, A324012 (self-conjugate).

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}]

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

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)

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

A086364 Triangle read by rows: S_D(n,k) = `Type D' Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 2, 2, 9, 4, 2, 27, 36, 10, 2, 65, 195, 140, 26, 2, 143, 840, 1180, 540, 76, 2, 301, 3171, 7735, 6510, 2142, 232, 2, 619, 11060, 43659, 59920, 34692, 8624, 764, 2, 1257, 36707, 223566, 467691, 423612, 180852, 35856, 2620, 2, 2535, 117960, 1071350

Offset: 0

James East, Sep 04 2003

Cf. A085483, A086365 (row sums).

A partition of {-n, ..., -1, 1, ..., n} into nonempty subsets X_1, ..., X_r is called `symmetric' if for each i -X_i = X_j for some j. S_D(n, k) is the number of such symmetric partitions whose induced partition on {1, ..., n} involves k nonempty subsets and none of the X_i are of the form {j, -j}.

cp's OEIS Frontend

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

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