Diametric problem for permutations with the Ulam metric (optimal anticodes)

Abstract

We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let $S_n$ denote the set of permutations on $n$ symbols, and for each $\sigma, \tau \in S_n$, define their Ulam distance as the number of distinct symbols that must be deleted from each until they are equal. We obtain a near-optimal upper bound on the size of the intersection of two balls in this space, and as a corollary, we prove that a set of diameter at most $k$ has size at most $2^{k + C k^{2/3}} n! / (n-k)!$, compared to the best known construction of size $n!/(n-k)!$. We also prove that sets of diameter $1$ have at most $n$ elements.