Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partial geodesics on symmetric groups endowed with breakpoint distance   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.

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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga