论文标题
得分序列的渐近数
The asymptotic number of score sequences
论文作者
论文摘要
图表上的比赛是其边缘的方向。得分序列以非降低顺序列出了阶段。 Winston和Kleitman（1983）和Kim and Pittel（2000）的作品表明，完整图上的得分序列的$ s_n $ $ k_n $满足$ s_n =θ（4^n/n^{5/2}）$。通过将$ s_n $的最新复发关系结合在erdős-ginzburg-ziv数字$ n_n $方面，与离散无限分布的极限理论相结合，我们观察到$ n^{5/2} s_n/4^n \ n_k/k4^k $。该限制与Takács（1986）猜想的$ s_n $的渐近学一致。我们还确定了强分序的渐近数，并表明随机得分序列中的不可约子谱的数量在分布中收敛到参数的移位负二项式$ r = 2 $和$ p = e^e^{ - λ} $。
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 $S_n=Θ(4^n/n^{5/2})$. 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 $n^{5/2}S_n/4^n\to e^λ/2\sqrtπ$, where $λ=\sum_{k=1}^\infty N_k/k4^k$. 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 $p=e^{-λ}$.
