Secant Tree Calculus

TL;DR

This paper develops a Tree Calculus to analyze the joint distribution of two statistics on secant trees, providing explicit generating functions and symmetry properties for their restricted set.

Contribution

It introduces a new Tree Calculus framework for secant trees, deriving explicit generating functions and difference equations for joint statistics.

Findings

Joint distribution satisfies two partial difference equations

Distribution restricted to {eoc-pom<= 1} is symmetric

Explicit three-variable generating function derived

Abstract

A true Tree Calculus is being developed to make a joint study of the two statistics "eoc" (end of minimal chain) and "pom" (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom<= 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.