Refined enumeration of $k$-plane trees and $k$-noncrossing trees
Combinatorics
2022-07-12 v2
Abstract
A -plane tree is a plane tree whose vertices are assigned labels between and in such a way that the sum of the labels along any edge is no greater than . These trees are known to be related to -ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for -noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to -ary trees.
Keywords
Cite
@article{arxiv.2205.01002,
title = {Refined enumeration of $k$-plane trees and $k$-noncrossing trees},
author = {Isaac Owino Okoth and Stephan Wagner},
journal= {arXiv preprint arXiv:2205.01002},
year = {2022}
}