2023/02/10 by Sergi Elizalde, Elizalde, Sergi
Agricultural and Biological Sciences · Computer Science · Environmental Science · #05A15 #05A16 #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #Ecology and Vegetation Dynamics Studies #FOS: Mathematics #Sensory Analysis and Statistical Methods
paper · pdf · doi:10.48550/arxiv.2302.05252
openalex publication_date 2023/02/10 · openalex created_date 2023/02/15 · openalex updated_date 2026/07/28
We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work of Panholzer and Prodinger, we apply this method to study the depth distributions of individual nodes in rooted binary trees, plane trees, noncrossing trees, and increasing trees; the height distributions of individual vertices and individual steps in Dyck paths; and the number of diagonals separating two fixed sides of a convex polygon in a triangulation or dissection. We obtain both exact and asymptotic results, which sometimes refine known formulas or provide combinatorial proofs of results from the probability literature.