具有断点距离的对称群上的部分测地线
Partial geodesics on symmetric groups endowed with breakpoint distance
摘要：
Jamshidpey等人在2014年的“Sets of medians in the non-geodesic pseudometric space of unsigned genomes with breakpoints”中引入了部分测地线的概念。本文研究了对称群$S_n$中随机一致独立选取的两个置换$ξ_1^{(n)}$和$ξ_2^{(n)}$之间的非平凡部分测地线上的点的密度，其中$S_n$被赋予断点距离.对于置换$π:= π_1 \ ... \ π_n$，任何无序对$\{π_i , π_{i+1}\}$，对于$i=1, ..., n-1$，称为$π$的邻接。$π$的所有邻接的集合由$\mathcal{A}_π$表示。用$id^{(n)}$表示单位置换，并且令$I_n$是$\mathcal A_{id^{(n)}}$的任意子集。我们把一个置换$π\in S_n$的所有邻接的集合分成四种类型，相对于$I_n$。然后，对于从$S_n$中随机一致选择的置换$ξ^{(n)}$，我们导出了$ξ^{(n)}$中每种类型的邻接的归一化数（除以$n$后）关于$I_n$（对于$I_n$的某些随机或确定性选择）的收敛定理，为$n\rightarrow \infty$。我们还看到了这个收敛定理的应用，找到适当的选择$I_n$。伪度量空间$(S,ρ)$中$u$和$v$的测地线点是$ρ(u,w)+ρ(w,v)=ρ(u,v)$的空间的点$w$。我们找到一个上限的排列$x\in S_n$的数量，其中存在至少一个非平凡的测地线点之间的$id^{(n)}$和$x$，远离两者。这部分验证了Haghighi和Sankoff在2012年的“Medians seek the corners，and other atturtures”中提出的猜想，即我们证明了，以很高的概率，不存在从$S_n$中随机一致独立地选择的两个排列$ξ_1^{(n)}$和$ξ_2^{(n)}$的断点中值，远离它们。
英文摘要：
The notion of partial geodesic was introduced by Jamshidpey et al. in "Sets of medians in the non-geodesic pseudometric space of unsigned genomes with breakpoints", 2014. In this paper, we study the density of points on non-trivial partial geodesics between two permutations $ξ_1^{(n)}$ and $ξ_2^{(n)}$ chosen uniformly and independently at random from the symmetric group $S_n$, where $S_n$ is endowed with the breakpoint distance. For a permutation $π:= π_1 \ ... \ π_n$, any unordered pair $\{π_i , π_{i+1}\}$, for $i=1, ..., n-1$, is called an adjacency of $π$. The set of all adjacencies of $π$ is denoted by $\mathcal{A}_π$. Denote by $id^{(n)}$ the identity permutation, and let $I_n$ be an arbitrary subset of $\mathcal A_{id^{(n)}}$. We classify the set of all adjacencies of a permutation $π\in S_n$ into four types, with respect to $I_n$. Then for a permutation $ξ^{(n)}$ chosen uniformly at random from $S_n$, we derive a convergence theorem for the normalized number (after dividing by $n$) of adjacencies of each type in $ξ^{(n)}$ with respect to $I_n$ (for some random or deterministic choices of $I_n$), as $n\rightarrow \infty$. We also see an application of this convergence theorem to find the appropriate choices of $I_n$. A geodesic point of $u$ and $v$ in a pseudometric space $(S,ρ)$ is a point $w$ of the space that $ρ(u,w)+ρ(w,v)=ρ(u,v)$. We find an upper bound for the number of permutations $x\in S_n$ for which there exists at least one non-trivial geodesic point between $id^{(n)}$ and $x$, far from both. This partially verifies the conjecture of Haghighi and Sankoff stated in "Medians seek the corners, and other conjectures", 2012, namely we prove that, with high probability, there is no breakpoint median of two permutations $ξ_1^{(n)}$ and $ξ_2^{(n)}$ chosen uniformly and independently at random from $S_n$, far from both of them.
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