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