The asymptotic number of score sequences

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 $S_n=\mathrm{\Theta }(4^n\mathrm{/}{n}^{5\mathrm{/}2})$. 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 $n^{5\mathrm{/}2}{S}_n\mathrm{/}{4}^n\to e^{\lambda }\mathrm{/}2\sqrt{\pi }$, where $\lambda ={\sum }_{k=1}^{\mathrm{\infty }}{N}_k\mathrm{/}k{4}^k$. 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 $p=e^{-\lambda }$.