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%I #14 Feb 16 2022 23:22:27
%S 1,0,1,1,4,3,15,16,75,89,428,571,2781,4060,20093,31697,159340,268791,
%T 1372163,2455804,12725447,24012697,126238060,249880687,1332071241,
%U 2754348360,14881206473,32029000641,175297058228,391548016475,2169832010759
%N Number of self-complementary set partitions of {1, ..., n} with no singletons.
%C 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}}.
%H Andrew Howroyd, <a href="/A324013/b324013.txt">Table of n, a(n) for n = 0..500</a>
%H David Callan, <a href="https://arxiv.org/abs/math/0508052">On conjugates for set partitions and integer compositions</a>, arXiv:math/0508052 [math.CO], 2005.
%F From _Andrew Howroyd_, Feb 16 2022: (Start)
%F a(2*n) = A086365(n-1) for n > 0.
%F a(2*n) = n!*[x^n] exp((exp(2*x) - 3)/2 - x + exp(x));
%F a(2*n+1) = n!*[x^n] (exp(x) - 1)*exp((exp(2*x) - 3)/2 - x + exp(x)).
%F (End)
%e The a(3) = 1 through a(6) = 15 self-complementary set partitions with no singletons:
%e {{123}} {{1234}} {{12345}} {{123456}}
%e {{12}{34}} {{135}{24}} {{123}{456}}
%e {{13}{24}} {{15}{234}} {{124}{356}}
%e {{14}{23}} {{1256}{34}}
%e {{1346}{25}}
%e {{135}{246}}
%e {{145}{236}}
%e {{16}{2345}}
%e {{12}{34}{56}}
%e {{13}{25}{46}}
%e {{14}{25}{36}}
%e {{15}{26}{34}}
%e {{16}{23}{45}}
%e {{16}{24}{35}}
%e {{16}{25}{34}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
%t Table[Select[sps[Range[n]],And[cmp[#]==Sort[#],Count[#,{_}]==0]&]//Length,{n,0,10}]
%o (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
%Y Cf. A000110, A000296, A080107 (self-complementary), A086365, A124323, A324012 (self-conjugate).
%K nonn
%O 0,5
%A _Gus Wiseman_, Feb 12 2019
%E Terms a(13) and beyond from _Andrew Howroyd_, Feb 16 2022
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