The asymptotic number of score sequences

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 $p=e^{-\lambda}$.