Making simple proofs simpler

TL;DR

This paper investigates the combinatorial properties of open partitions in specific tree structures, focusing on the count of such partitions in a tree composed of two chains sharing a root.

Contribution

It provides a detailed analysis of the number of open partitions in a particular class of trees, extending understanding of their combinatorial structure.

Findings

Derived formulas for NP(n), the number of open partitions in the given tree structure.

Established connections between open partitions and combinatorial tree properties.

Enhanced the theoretical framework for analyzing partitions in rooted trees.

Abstract

An open partition \pi{} [Cod09a, Cod09b] of a tree T is a partition of the vertices of T with the property that, for each block B of \pi, the upset of B is a union of blocks of \pi. This paper deals with the number, NP(n), of open partitions of the tree, V_n, made of two chains with n points each, that share the root.