  We study a subset of permutations, where entries are restricted to having the
same remainder as the index, modulo some integer $k \geq 2$. We show that when
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, avoiding two patterns of length 3. This is analogous to the
systematic study by Simion and Schmidt, of permutations avoiding two patterns
of length 3.
