Generalized Stirling permutations, families of increasing trees and urn models

Bona [6] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [14]. Recently, Janson [18] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of [6], [18], and relate them with certain families of generalized plane recursive trees, and also (k + 1)-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) P´olya urn models using various methods.

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Svante Janson, Markus Kuba, Alois Panholzer
• Keywords: Certain Families, JCT 2011, Permutations, Plane Recursive Trees