cp's OEIS Frontend

A097236 Number of hierarchical orderings ("societies") with at least 2 elements ("individuals") on each level for n labeled elements.

%I A097236 #24 Dec 12 2021 22:29:49
%S A097236 0,1,1,10,31,271,1534,14393,117653,1253524,13140557,160679069,
%T A097236 2026451948,28278471729,413532314433,6516434058758,107958571213579,
%U A097236 1899723866781859,35092386753388698,682552407940860353,13893916425860413469,296049402365644855888
%N A097236 Number of hierarchical orderings ("societies") with at least 2 elements ("individuals") on each level for n labeled elements.
%H A097236 Alois P. Heinz, <a href="/A097236/b097236.txt">Table of n, a(n) for n = 1..175</a>
%H A097236 N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%F A097236 E.g.f.: exp(-(-exp(z)+1+z)/(2-exp(z)+z)).
%F A097236 a(n) ~ exp(1/(2*(c-2)) + 1/(2*(c-1)^2) + 2*sqrt(n/((c-2)*(c-1))) - n - 1) * n^(n-1/4) / (sqrt(2) * (c-1)^(1/4) * (c-2)^(n+1/4)), where c = -LambertW(-1, -exp(-2)) = A226572 = 3.14619322062... . - _Vaclav Kotesovec_, Sep 08 2014
%e A097236 a(4) = 10. Let : denote the partition of n labeled individuals 1,2,3,4 into x=2 sets (i.e. "societies"). E.g., in 12:34 one has a single society with members 1 and 2 and a further single society with members 3 and 4. Let | denote a higher level (within a single society), e.g., in 1|2 the individual 2 is one level up with respect to individual 1. The order of individuals on a level is insignificant, e.g. 12|34 is equivalent to 21|43.
%e A097236 For n = 4 and x = 2 one has 1234; 12:34; 13:24; 14:23; 12|34; 31|42; 43|21; 24|13; 21|34; 43|12; which gives 10 different hierarchical societies with at least 2 labeled individuals per level.
%p A097236 with(combstruct); SetSeqSetxL:=[T,{T=Set(S), S=Sequence(U,card>=1), U=Set(Z,card >= 2)},labeled];
%p A097236 # where x is an integer 1, 2, 3,... ; x=2 gives 2 individuals per level.
%p A097236 seq (count (SetSeqSetxL,size=j), j=1..20);
%t A097236 terms = 22;
%t A097236 CoefficientList[ Exp[-(-Exp[z]+1+z)/(2-Exp[z]+z)] + O[z]^(terms+1), z] *  Range[0, terms]! // Rest (* _Jean-François Alcover_, Aug 06 2018 *)
%Y A097236 Cf. A075729, A097237, A226572.
%K A097236 nonn
%O A097236 1,4
%A A097236 _Thomas Wieder_, Aug 02 2004