  Comtet introduced the notion of indecomposable permutations in 1972. A
permutation is indecomposable if and only if it has no proper prefix which is
itself a permutation. Indecomposable permutations were studied in the
literature in various contexts. In particular, this notion has been proven to
be useful in obtaining non-trivial enumeration and equidistribution results on
permutations.
  In this paper, we give a complete classification of indecomposable
permutations avoiding a classical pattern of length 3 or 4, and of
indecomposable permutations avoiding a non-consecutive vincular pattern of
length 3. Further, we provide a recursive formula for enumerating $12\cdots
k$-avoiding indecomposable permutations for $k\geq 3$. Several of our results
involve the descent statistic. We also provide a bijective proof of a fact
relevant to our studies.
