The asymptotic number of score sequences

TL;DR

This paper derives the asymptotic behavior of the number of score sequences in tournaments, linking it to Erdős–Ginzburg–Ziv numbers and revealing distributional properties of subscores.

Contribution

It combines recurrence relations and limit theory to establish the asymptotics of score sequences and their irreducible components, confirming a conjecture by Takács.

Findings

Asymptotic formula for total score sequences $S_n$ matches conjectured behavior.

Number of strong score sequences is asymptotically characterized.

Distribution of irreducible subscores converges to a shifted negative binomial.

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 $S_n=\Theta(4^n/n^{5/2})$. By combining a recent recurrence relation for $S_n$ in terms of the Erd\H{o}s--Ginzburg--Ziv numbers $N_n$ with the limit theory for discrete infinitely divisible distributions, we observe that $n^{5/2}S_n/4^n\to e^\lambda/2\sqrt{\pi}$, where $\lambda=\sum_{k=1}^\infty N_k/k4^k$. This limit agrees numerically with the asymptotics of $S_n$ conjectured by Tak\'acs (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 $p=e^{-\lambda}$.