A097237 - OEIS

%I #21 Dec 12 2021 22:30:21

%S 0,2,12,86,780,8462,106092,1507046,23905740,418581662,8014481772,

%T 166501716086,3728936827980,89530481995502,2293539506425452,

%U 62429371709206406,1799021068567370700,54707449240102350782,1750530594833378049132,58787407236482804618006

%N Number of hierarchical orderings ("societies") of n labeled elements ("individuals") with at least two occupied levels.

%H Alois P. Heinz, <a href="/A097237/b097237.txt">Table of n, a(n) for n = 1..150</a>

%H 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 E.g.f.: exp(-(exp(z)^2-2*exp(z)+1)/(-2+exp(z))).

%F a(n) ~ exp(sqrt(2*n/log(2)) + 1/(4*log(2)) - n - 7/4) * n^(n-1/4) / (2^(3/4) * log(2)^(n+1/4)). - _Vaclav Kotesovec_, Sep 13 2014

%e a(3) = 12. 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. For n = 3 and x = 2 one has 12|3; 23|1; 13|2; 1|23; 2|13; 3|12; 1|2|3; 2|3|1; 3|1|2; 1|3|2; 3|2|1; 2|1|3; which gives 12 different societies with at least 2 occupied levels.

%p with(combstruct); SetSeq2SetL:=[T,{T=Set(S), S=Sequence(U,card>=2), U=Set(Z,card >= 1)},labeled];

%p # where x is an integer 1, 2, 3,... ; x=2 gives 2 levels per society.

%p seq (count (SetSeq2SetL,size=j), j=1..12);

%t Rest[CoefficientList[Series[E^((2*E^x-E^(2*x)-1) / (E^x-2)), {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Sep 13 2014 *)

%Y Cf. A075729, A097236.

%K nonn

%O 1,2

%A _Thomas Wieder_, Aug 02 2004