The asymptotic number of score sequences (Q6081400)

From the author's introduction: ``A tournament on a graph \(G\) is an orientation of its edges. Informally, each vertex is a team and each pair of vertices joined by an edge plays a game. Afterwards, the edge is directed towards the winner. The score sequence lists the total number of wins by each team in non-decreasing order.'' Let \(S_n\) denote the number of score sequences on the complete graph on \(n\) vertices and let \(N_n\) denote the number of subsets of size \(n\) of the set \(\{1, 2, \dots , 2n - 1\}\) whose sum is a multiple of \(n\). In this paper, the author confirms a conjecture of \textit{L. Takács} [J. Stat. Plann. Inference 14, 123--142 (1986; Zbl 0616.60016)] for the asymptotic of \(S_n\) as \(n \to\infty\). He shows that \(n^{5/2}S_n/4^n \to e^{\lambda}/2 \sqrt{\pi}\approx 0.392\), where \(\lambda=\sum_{k=1}^{\infty}N_k/k4^k\). He also identifies the asymptotic number of strong score sequences, that is, score sequences of a strongly connected tournament (in which each pair of vertices is in a directed cycle), and observes that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial random variable with parameters 2 and \(e^{-\lambda}\). The method of proof combines a recurrence relation for \(S_n\) in terms of the numbers \(N_n\) with the limit theory for discrete infinitely divisible distributions.