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%I #23 May 30 2019 16:58:02
%S 1,6,53,619,8972,155067,3109269,70893872,1810283331,51151579619,
%T 1583934062306,53322541667501,1938521128765093,75673000809822670,
%U 3156390306304019025,140076451219218605087,6589244960448222899044,327461842184597424792623,17141751726301435708168665
%N Number of hierarchical orderings for n labeled elements with 2 possible classes A and B for levels l>=2. Labeled analog of A104460.
%H Alois P. Heinz, <a href="/A109092/b109092.txt">Table of n, a(n) for n = 1..140</a>
%H Robert Gill, <a href="https://doi.org/10.1016/S0012-365X(97)00187-8">The number of elements in a generalized partition semilattice</a>, Discrete mathematics 186.1-3 (1998): 125-134. See Example 2.
%H Norihiro Nakashima, Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%H N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
%F G.f.: exp(-(exp(z)-1)/(-3+2*exp(z))).
%e Let | denote a separator among different hierarchies of the hierarchical ordering. Let : denote a separator between levels in a hierarchy.
%e Furthermore, let a[1], a[2],... denote labeled elements.
%e An element a[i] will be written as a[i,A] if it falls into class A and as a[i,B] if it falls into class B. Note that at level l=1 no classes appear.
%e Then a(2) = 6 because a[1]a[2], a[1]|a[2], a[1]:a[2,A], a[2]:a[1,A], a[1]:a[2,B], a[2]:a[1,B].
%p with(combstruct): A109092 := [T, {T=Set(Sequence(S,card>=1)), S=Sequence(U,card>=1), U=Set(Z,card>=1)},labeled]; seq(count(A109092, size=j), j=1..20);
%t With[{nn=20},CoefficientList[Series[Exp[-(Exp[x]-1)/(-3+2Exp[x])],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Sep 16 2016 *)
%Y Cf. A075729, A104460.
%K nonn
%O 1,2
%A _Thomas Wieder_, Jun 18 2005
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