TY  - JOUR
A1  - Kuba, Markus
A1  - Varvak, Anna
T1  - On Path diagrams and Stirling permutations
JF  - Séminaire Lotharingien de Combinatoire
N2  - A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths.

The aim of this article is to extend the notion of local types from permutations to k-Stirling permutations (also known as k-multipermutations). We establish a bijection of these local types to node types of (k+1)-ary increasing trees. We present a branched continued fraction representation of the generating function of these local types through a bijection with path diagrams induced by Łukasiewicz paths, generalizing the results from permutations to arbitrary k-Stirling permutations.

We further show that the generating function of ordinary Stirling permutation has at least three branched continued fraction representations, using correspondences between non-standard increasing trees, k-Stirling permutations and path diagrams.
KW  - Kettenbruch
KW  - Formale Potenzreihe
KW  - Continued fractions
KW  - Łukasiewicz paths
KW  - Path diagrams
KW  - Stirling permutations
KW  - Multipermutations
KW  - Increasing trees
Y1  - 2021
IS  - B82c (2021)
ER  - 
TY  - JOUR
A1  - Kuba, Markus
A1  - Panholzer, Alois
T1  - Tree evolution processes for bucket increasing trees
JF  - Discrete Mathematics
N2  - Bucket increasing trees are multilabelled generalizations of increasing trees, where each non-leaf node carries b labels, with a fixed integer. We provide a fundamental result, giving a complete characterization of all families of bucket increasing trees that can be generated by a tree evolution process. We also provide several equivalent properties, complementing and extending earlier results for ordinary increasing trees to bucket trees. Additionally, we state second order results for the number of descendants of label j, again extending earlier results in the literature.
KW  - Increasing trees
KW  - multilabelled trees
KW  - tree evolution processes
Y1  - 
U6  - http://dx.doi.org/https://doi.org/10.1016/j.disc.2023.113443
VL  - Vol. 346
IS  - Issue 7
SP  - 113443
ER  -