Tournament score sequences, Erdős-Ginzburg-Ziv numbers, and the Lévy-Khintchine method

TL;DR

The paper develops a concise probabilistic approach to the asymptotic enumeration of score sequences $S_n$ in tournaments, revealing a deep link to Erdős–Ginzburg–Ziv numbers $N_n$ via the Lévy–Khintchine method. By leveraging the lattice-path representation of score sequences and Kleitman’s cyclic-shift ideas, the authors derive a simple proof that $S_n^*=N_n$ and use renewal-sequence theory to connect $S_n$ to $N_n$, ultimately confirming Moser's conjecture that $S_n\sim C 4^n/n^{5/2}$ with $C$ expressed in terms of $N_n$. The main innovations are the geometric diamond-area interpretation of bridges, the renewal-structure framework for analyzing cyclic shifts, and a clean bijection that ties the two counting problems together. The results illustrate the power of the Lévy–Khintchine paradigm in combinatorial asymptotics and suggest broader applicability to renewal-like structures in discrete geometry and permutation-based polytopes.

Abstract

We give a short proof of a recent result of Claesson, Dukes, Franklín and Stefánsson, connecting the number $S_n$ of score sequences and the Erdős-Ginzburg-Ziv numbers $N_n$ from additive number theory. Our proof utilizes the lattice path representation of score sequences by Erdős 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 Lévy-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 $N_n$.

Figures (3)

• Figure 1: The permutahedron $\Pi_3\subset\mathbb{R}^4$ (projected into $\mathbb{R}^3$) is the convex hull of 0123 and its permutations. Its non-decreasing lattice points 0123, 0222, 1113 and 1122 are the $S_4=4$ score sequences.
• Figure 2: A bridge $B$ (solid) of length $10=2\cdot 5$, with down steps at times $3$, $4$, $5$, $7$ and $8$. There are 3 diamonds (red) above and 1 diamond (blue) below the sawtooth bridge (dotted), so its diamond area is $a(B)=3-1=2$. Equivalently, in terms of its down steps, $a(B)=-5^2+(3+4+5+7+8)=2$.
• Figure 3: The $S_4=4$ bridges $B$ (at left) associated with the score sequences 0123, 0222, 1113 and 1122 and the $2N_4=4+14=18$ bridges $B'$ (at left and right) with $a(B)\equiv0$ mod $4$. To obtain a bijective correspondence, we cyclically shift bridges $B$ by some $m$ (black dots) less than the length $2\ell$ (white dots) of their first irreducible parts.

Theorems & Definitions (3)

• Theorem 1
• Lemma 2
• proof