Abstract
Let fnbe a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon Pnof n sides. Suppose Tnis a random triangulation, sampled uniformly out of all possible triangulations of Pn. We study the sum of weights of triangles in Tnand give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of fnin which we obtain explicit forms of generating functions for the sum of the weights. For example, among other things, we give new proofs for already known results such as the degree of a fixed vertex and the number of ears in Tn, as well as, provide new results on the number of “blue” angles and refined information on the distribution of angles at a fixed vertex. We note that our approach is systematic and can be applied to many other new examples while generalizing the existing results.
Original language | English |
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Pages (from-to) | 389-412 |
Number of pages | 24 |
Journal | Journal of Combinatorics |
Volume | 12 |
Issue number | 3 |
DOIs | |
State | Published - 2021 |
Bibliographical note
Publisher Copyright:© 2021, Journal of Combinatorics. All rights reserved.
Keywords
- Convex polygon
- Random triangulation
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics