A097391 The number of hierarchies of n labeled elements with at least one subhierarchy composed of exactly 2 levels and no subhierarchy with more than 2 levels.

0, 1, 3, 8, 17, 37, 71, 138, 252, 458

Offset: 1

Author

Thomas Wieder, Aug 13 2004

Keywords

• nonn
• more

Comments

Conjectured to be (A378854) /3. - Wouter Meeussen, Feb 10 2025

Examples

			Let : denote the separation between two subhierarchies, e.g. 2:3 are two subhierarchies where subhierarchy s=1 contains two elements and subhierarchy s=2 contains three elements. Let | denote the separation between two levels, e.g. 2|2|1 is a hierarchy composed of three levels with two elements on levels l=1 and l=2 and one element on level l=3. For n=5 one has a(5) = 17 hierarchies where at least one subhierarchy has exactly 2 levels (and no level l > 2 is allowed):
4|1; 1|4; 3|2; 2|3; 2|2:1; 2|1:2; 1|2:2; 2|1:1|1; 1|2:1|1; 2|1:1:1; 1|2:1:1; 1|1:1:1:1; 1|3:1; 3|1:1; 1|1:2:1; 1|1:1|1:1; 1|1:3.
		

Links

• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Crossrefs

Cf. A034691, A097237, A097392, A000041, A378854.