Median inverse problem and approximating the number of $k$-median inverses of a permutation

We introduce the “Median Inverse Problem” for metric spaces. In particular, having a permutation $\pi$ in the symmetric group $S_n$ (endowed with the breakpoint distance), we study the set of all $k$-subsets $\{x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k\}\subset S_n$ for which $\pi$ is a breakpoint median. The set of all $k$-tuples $(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k)$ with this property is called the $k$-median inverse of $\pi$. Finding an upper bound for the cardinality of this set, we provide an asymptotic upper bound for the probability that $\pi$ is a breakpoint median of $k$ permutations ${\xi }_1^{(n)},\mathrm{.}\mathrm{.}\mathrm{.},{\xi }_k^{(n)}$ chosen uniformly and independently at random from $S_n$.