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We introduce the "Median Inverse Problem" for metric spaces. In particular,having a permutation $\pi$ in the symmetric group $S_n$ (endowed with thebreakpoint distance), we study the set of all $k$-subsets$\{x_1,...,x_k\}\subset S_n$ for which $\pi$ is a breakpoint median. The set ofall $k$-tuples $(x_1,...,x_k)$ with this property is called the $k$-medianinverse of $\pi$. Finding an upper bound for the cardinality of this set, weprovide an asymptotic upper bound for the probability that $\pi$ is abreakpoint median of $k$ permutations $\xi_1^{(n)},...,\xi_k^{(n)}$ chosenuniformly and independently at random from $S_n$.

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