All binomial identities are orderable

TL;DR

This paper proves that all binomial identities can be ordered and explores their applications in combinatorics and distributed computing, introducing the concept of fundamental binomial identities and identifying new non-existence cases.

Contribution

It establishes the orderability of all binomial identities and introduces the notion of fundamental binomial identities, expanding understanding in combinatorics and distributed computing.

Findings

All binomial identities are orderable.

Introduces the concept of fundamental binomial identities.

Identifies new values where no fundamental binomial identity exists.

Abstract

The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on the round complexity of the weak symmetry breaking task. Furthermore, we introduce the notion of a fundamental binomial identity and find an infinite family of values, other than the prime powers, for which no fundamental binomial identity can exist.