The Asymptotic Number of Score Sequences

Abstract

AbstractA 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 $$S_n$$ S n of score sequences on the complete graph $$K_n$$ K n satisfies $$S_n=\Theta (4^n/n^{5/2})$$ S n = Θ ( 4 n / n 5 / 2 ) . By combining a recent recurrence relation for $$S_n$$ S n in terms of the Erdős–Ginzburg–Ziv numbers $$N_n$$ N n with the limit theory for discrete infinitely divisible distributions, we observe that $$n^{5/2}S_n/4^n\rightarrow e^\lambda /2\sqrt{\pi }$$ n 5 / 2 S n / 4 n → e λ / 2 π , where $$\lambda =\sum _{k=1}^\infty N_k/k4^k$$ λ = ∑ k = 1 ∞ N k / k 4 k . This limit agrees numerically with the asymptotics of $$S_n$$ 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 $$r=2$$ r = 2 and $$p=e^{-\lambda }$$ p = e - λ .