Pattern frequency sequences and internal zeros

Let q be a pattern and let S-n,S-q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (S-n,S-q(c))(cgreater than or equal to0) 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 I = 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. (C) 2002 Elsevier Science (USA).