The asymptotic number of score sequences

Brett Kolesnik

Published 2022-09-27Version 1

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 $S_n=\Theta(4^n/n^{5/2})$. In this note, we observe that by combining recent combinatorial developments related to score sequences with the limit theory for discrete infinitely divisible distributions it follows that $n^{5/2}S_n/4^n\to 0.392\ldots$, as conjectured by Tak\'acs (1986).