Tournament score sequences, Erd\H{o}s-Ginzburg-Ziv numbers, and the L\'evy-Khintchine method

TL;DR

This paper presents a concise proof linking score sequences and Erd ext{"o}s-Ginzburg-Ziv numbers using lattice path representations and probability theory, confirming Moser's conjecture on their asymptotic behavior.

Contribution

It introduces a novel proof connecting score sequences with additive number theory via the Lévy-Khintchine formula, simplifying previous results and confirming conjectures.

Findings

Established a connection between score sequences and Erd ext{"o}s-Ginzburg-Ziv numbers

Provided a short proof of Moser's conjecture on asymptotic behavior

Utilized lattice path representation and probability theory techniques

Abstract

We give a short proof of a recent result of Claesson, Dukes, Frankl\'in and Stef\'ansson, connecting the number $S_n$ of score sequences and the Erd\H{o}s-Ginzburg-Ziv numbers $N_n$ from additive number theory. Our proof utilizes the lattice path representation of score sequences by Erd\H{o}s and Moser, and remarks by Kleitman added to an article of Moser regarding cyclic shifts of such paths. The connection between $S_n$ and $N_n$ is an instance of the L\'evy-Khintchine formula from probability theory. We highlight the utility of such formulas, by giving a short proof of Moser's conjecture that $S_n\sim C4^n/n^{5/2}$, where $C$ is described in terms of $N_n$.