The asymptotic number of score sequences

TL;DR

This work derives the precise asymptotics for the number of score sequences $S_n$ on the complete graph, confirming Takács' conjecture by establishing $\lim_{n\to\infty} \frac{n^{5/2}S_n}{4^n}=\frac{e^{\lambda}}{2\sqrt{\pi}}$ with $\lambda=\sum_{k\ge1}\frac{N_k}{k4^k}$. It couples a new recurrence linking $S_n$ to the Erdős–Ginzburg–Ziv numbers $N_n$ with discrete infinite-divisibility limit theory to transfer asymptotics, and leverages renewal-analytic arguments to characterize irreducible subscores and strong score sequences. The results show that a majority of score sequences are strong in the limit and that the number of irreducible subscores converges in distribution to a shifted negative-binomial with parameters $(r=2, p=e^{-\\lambda})$, providing a rich probabilistic interpretation and connecting combinatorics with renewal theory. Overall, the paper advances the understanding of score-sequence combinatorics and establishes a bridge between enumerative structures and infinite-divisibility frameworks.

Abstract

A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (1983) and Kim and Pittel (2000) showed that the number $S_n$ of score sequences on the complete graph $K_n$ satisfies $S_n=Θ(4^n/n^{5/2})$. By combining a recent recurrence relation for $S_n$ in terms of the Erdős--Ginzburg--Ziv numbers $N_n$ with the limit theory for discrete infinitely divisible distributions, we observe that $n^{5/2}S_n/4^n\to e^λ/2\sqrtπ$, where $λ=\sum_{k=1}^\infty N_k/k4^k$. This limit agrees numerically with the asymptotics of $S_n$ conjectured by Takács (1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters $r=2$ and $p=e^{-λ}$.

Figures (3)

• Figure 1: The permutahedron $\Pi_3$.
• Figure 2: The limiting subscore distribution, negative binomial with parameters $r=2$ and $p=e^{-\lambda}$.
• Figure 3: The infinitely divisible tournament distribution, $p_n=e^{-\lambda}S_n/4^n$.

Theorems & Definitions (19)

• Theorem 1
• Corollary 2
• Corollary 3
• Corollary 4
• Corollary 5
• proof : Proof of \ref{['T_main']}
• Theorem 6: EH82
• proof : Proof of \ref{['C_main']}
• Proposition 7
• Lemma 8: HJ78