On Path diagrams and Stirling permutations

Any ordinary permutation τ ∈ Sn of size n, written as a word τ = τ1 . . . τn, can be locally classified according to the relative order of τj to its neighbours. This gives rise to four local order types called peaks (or maxima), valleys (or minima), double rises and double falls. By the correspondence between permutations and binary increasing trees the classification of permutations according to local types corresponds to a classification of binary increasing trees according to nodes types. Moreover, by the bijection between permutations, binary increasing trees and suitably defined path diagrams one can obtain continued fraction representations of the ordinary generating function of local types. The aim of this work is to introduce the notion of local types in k-Stirling permutations, to relate these local types with nodes types in (k + 1)-ary increasing trees and to obtain a bijection with suitably defined path diagrams. Furthermore, we also present a path diagram representation of a related tree family called plane-oriented recursive trees, and the discuss the relation with ternary increasing trees.