A permutation τ in the symmetric group Sj is minimally overlapping if any two consecutive occurrences of τ in a permutation σ can share at most one element. Bóna \cite{B} showed that the proportion of minimal overlapping patterns in Sj is at least 3−e. Given a permutation σ, we let Des(σ) denote the set of descents of σ. We study the class of permutations σ∈Skn whose descent set is contained in the set {k,2k,…(n−1)k}. For example, up-down permutations in S2n are the set of permutations whose descent equal σ such that Des(σ)={2,4,…,2n−2}. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches 1 as k goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape (nk).