We study a subset of permutations where entries are restricted to having the same remainder as the index, modulo some integer k ≥ 2. We show that by also imposing the classical 132- or 213-avoidance restriction on the permutations, we recover the Fuss–Catalan numbers and some special cases of the Raney numbers. Surprisingly, an analogous statement also holds when we impose the mod k restriction on a Catalan family of subexcedant functions. Finally, we completely enumerate all combinations of mod-k-alternating permutations that avoid two patterns of length 3. This is analogous to the systematic study by Simion and Schmidt, of permutations avoiding two patterns of length 3.