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 (J Comb Theory Ser A 35(2):208–230, 1983) and Kim and Pittel (J Comb Theory Ser A 92(2):197–206, 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 Erdős–Ginzburg–Ziv numbers N_n with the limit theory for discrete infinitely divisible distributions, we observe that n^{5/2}S_n/4^nrightarrow e^lambda /2sqrt{pi }, where lambda =sum _{k=1}^infty N_k/k4^k. This limit agrees numerically with the asymptotics of S_n conjectured by Takács (J Stat Plan Inference 14(1):123–142, 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 }.