论文标题
在Mallows Distributions和Penstry $ p $降低的发行版中，连续数字的聚集
Clustering of consecutive numbers in permutations under Mallows distributions and super-clustering under general $p$-shifted distributions
论文作者
论文摘要
令$ a^{（n）} _ {l; k} \ subset s_n $表示$ [n] $的排列集，其中一组$ l $ conuctection $ \ {k，k+1，\ cdots，k+cdots，k+l-1 \} $出现在连续的位置中。在$ s_n $上的均匀概率度量下，一个$ p_n（a^{（n）} _ {l; k}）\ sim \ sim \ frac {l！} {n^{l-1}} $ as $ n \ as $ n \ to \ infty $。在本文的一部分中，我们考虑了在绿色分布下连续数字聚类的可能性$ p_n^q $，$ q> 0 $。由于双重性，因此考虑$ q \ in（0,1）$就足够了。我们表明，对于$ q_n = 1- \ frac c {n^α} $，带有$ c> 0 $和$α\ in（0,1）$，$ p_n^q（a^{（n）} _ {l; k_n; k_n}）$ $ \ {k_n \} _ {n = 1}^\ infty $。因此，让$ n^{（n）} _ l = \ sum_ {k = 1}^{n-l+1} 1_ {a^{（n）} _ {l; k}} $表示$ l $ $ l $的数字的集数，在连续的位置中出现在连续的位置中，我们已经连续\ lim at equient \ nequation \ equisation \ equient \ equient \ equienty* e_n^{q_n} n^{（n）} _ l = \ begin {case} \ infty，\ \ text {if} \ l <\ frac {1+α}α; \\ 0，\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ text {if} \ end {cases}。 \ end {等式*}我们还考虑$α= 1 $和$α> 1 $的情况。在本文的另一部分中，我们考虑了一般$ p $降低的发行版，其中摩洛斯分布是一种特殊情况。我们明确计算数量$ \ lim_ {l \ to \ infty} \ liminf_ {n \ to \ infty} p_n^q（a^{（n）} _ {l; k_n}）= \ lim_ {l \ to \ infty} \ limsup_ {n \ to \ infty} p_n^q（a^{（n）} _ {l; k_n}）$在$ p $ -distribution方面。当此数量为正时，我们说会发生超集群。特别是，使用参数$ q \ neq1 $的绿色分布进行超级群集。我们还给出了$ p $降低的分布的新特征。
Let $A^{(n)}_{l;k}\subset S_n$ denote the set of permutations of $[n]$ for which the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions. Under the uniformly probability measure $P_n$ on $S_n$, one has $P_n(A^{(n)}_{l;k})\sim\frac{l!}{n^{l-1}}$ as $n\to\infty$. In one part of this paper we consider the probability of clustering of consecutive numbers under Mallows distributions $P_n^q$, $q>0$. Because of a duality, it suffices to consider $q\in(0,1)$. We show that for $q_n=1-\frac c{n^α}$, with $c>0$ and $α\in(0,1)$, $P_n^q(A^{(n)}_{l;k_n})$ is on the order $\frac1{n^{α(l-1)}}$, uniformly over all sequences $\{k_n\}_{n=1}^\infty$. Thus, letting $N^{(n)}_l=\sum_{k=1}^{n-l+1}1_{A^{(n)}_{l;k}}$ denote the number of sets of $l$ consecutive numbers appearing in sets of consecutive positions, we have \begin{equation*} \lim_{n\to\infty} E_n^{q_n}N^{(n)}_l = \begin{cases}\infty,\ \text{if}\ l<\frac{1+α}α;\\ 0,\ \text{if} \ l>\frac{1+α}α. \end{cases}. \end{equation*} We also consider the cases $α=1$ and $α>1$. In the other part of the paper we consider general $p$-shifted distributions, of which the Mallows distribution is a particular case. We calculate explicitly the quantity $\lim_{l\to\infty} \liminf_{n\to\infty}P_n^q(A^{(n)}_{l;k_n}) = \lim_{l\to\infty}\limsup_{n\to\infty}P_n^q(A^{(n)}_{l;k_n})$ in terms of the $p$-distribution. When this quantity is positive, we say that super-clustering occurs. In particular, super-clustering occurs for the Mallows distribution with parameter $q\neq1$. We also give a new characterization of $p$-shifted distributions.
