Ascending Runs in Cayley Trees and Mappings
Year: 2015
Type: preprint
Abstract: We generalize the concept of ascending runs from permutations to Cayley trees and mappings. A combinatorial decomposition of the corresponding functional digraph allows us to obtain exact enumeration formulae that show an interesting connection to the Stirling numbers of the second kind. Using a bijective proof of Cayley's formula, we can show why the numbers of trees with a given number of ascending runs are directly linked to the corresponding numbers for mappings. Moreover, analytic tools allow a characterization of the limiting distribution of the random variable counting the number of ascending runs in a random mapping. As for permutations, this is a normal distribution with linear mean and variance.
Source: arXiv (Cornell University)
Authors: Marie-Louise Bruner, Alois Panholzer
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2201186759