Metrics on permutations with the same descent set

Let Sₙ be the symmetric group on the set [n]:=\1,2,…,n\. Given a permutation σ=σ₁σ₂ ⋯ σₙ ∈ Sₙ, we say it has a descent at index i if σᵢ>σᵢ₊₁. Let D(σ) be the set of all descents of σ and define D(S;n)=\σ∈ Sₙ\, | \,D(σ)=S\. We study the Hamming metric and _∞-metric on the sets D(S;n) for all possible nonempty S⊂[n-1] to determine the maximum possible value that these metrics can achieve when restricted to these subsets.