Partial geodesics on symmetric groups endowed with breakpoint distance

TL;DR

This paper investigates the structure of partial geodesics in the symmetric group with breakpoint distance, deriving convergence results for adjacency types and providing bounds on permutations with non-trivial geodesic points, supporting a conjecture about breakpoint medians.

Contribution

It introduces a convergence theorem for adjacency types in random permutations and bounds the number of permutations with non-trivial geodesic points, advancing understanding of geodesic structures in breakpoint metrics.

Findings

Convergence of adjacency type proportions in large random permutations.

Upper bounds on permutations with non-trivial geodesic points.

Support for the conjecture that random permutations lack distant breakpoint medians.

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.