Pattern Frequency Sequences and Internal Zeros

Let q be a pattern and let S"n","q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (S"n","q(c))"c">="0 has internal zeros. If q is a monotone pattern it turns out that, except for q=12 or 21, the nontrivial sequences (those where n is at least the length of q) always have internal zeros. For the pattern q=1(l+1)l...2 there are infinitely many sequences which contain internal zeros and when l=2 there are also infinitely many which do not. In the latter case, the only possible places for internal zeros are the next-to-last or the second-to-last positions. Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most 3.