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

0, 1, 1, 10, 31, 271, 1534, 14393, 117653, 1253524, 13140557, 160679069, 2026451948, 28278471729, 413532314433, 6516434058758, 107958571213579, 1899723866781859, 35092386753388698, 682552407940860353, 13893916425860413469, 296049402365644855888

Offset: 1

Author

Thomas Wieder, Aug 02 2004

Keywords

• nonn

Examples

			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.
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..175
• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Crossrefs

Cf. A075729, A097237, A226572.

Programs

Maple

with(combstruct); SetSeqSetxL:=[T,{T=Set(S), S=Sequence(U,card>=1), U=Set(Z,card >= 2)},labeled];
# where x is an integer 1, 2, 3,... ; x=2 gives 2 individuals per level.
seq (count (SetSeqSetxL,size=j), j=1..20);

Mathematica

terms = 22;
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 *)

Formula

E.g.f.: exp(-(-exp(z)+1+z)/(2-exp(z)+z)).

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