Partial geodesics on symmetric groups endowed with breakpoint distance

Abstract

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 $\xi_1^{(n)}$ and $\xi_2^{(n)}$ chosen uniformly and independently at random from the symmetric group $S_n$, where $S_n$ is endowed with the breakpoint distance. For a permutation $\pi := \pi_1 \ ... \ \pi_n$, any unordered pair $\{\pi_i , \pi_{i+1}\}$, for $i=1, ..., n-1$, is called an adjacency of $\pi$. The set of all adjacencies of $\pi$ is denoted by $\mathcal{A}_\pi$. 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 $\pi \in S_n$ into four types, with respect to $I_n$. Then for a permutation $\xi^{(n)}$ 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 $\xi^{(n)}$ 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 $(S,\rho)$ is a point $w$ of the space that $\rho(u,w)+\rho(w,v)=\rho(u,v)$. 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 $\xi_1^{(n)}$ and $\xi_2^{(n)}$ chosen uniformly and independently at random from $S_n$, far from both of them.