Metrics on permutations with the same descent set

Let Sₙ be the symmetric group on the set n: =\1, 2, , n\. Given a permutation =₁₂ ₙ Sₙ, we say it has a descent at index i if ᵢ>₈+₁. Let D () be the set of all descents of and define D (S;n) =\ Sₙ\, | \, D () =S\. We study the Hamming metric and _-metric on the sets D (S;n) for all possible nonempty S-1 to determine the maximum possible value that these metrics can achieve when restricted to these subsets.