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.

设\(q\)是一个模式，令\(S_{n,q}(c)\)为恰好有\(c\)个\(q\)副本的\(n\)排列的个数。我们研究序列\((S_{n,q}(c))_{c\geq0}\)何时有内部零点。如果\(q\)是一个单调模式，结果表明，除了\(q = 12\)或\(21\)之外，非平凡序列（即\(n\)至少是\(q\)的长度的那些序列）总是有内部零点。对于模式\(q = 1(l + 1)l\cdots2\)，有无限多个序列包含内部零点，并且当\(l = 2\)时，也有无限多个序列不包含内部零点。在后一种情况下，内部零点唯一可能的位置是倒数第二个或倒数第三个位置。注意，通过对称性，这完全确定了所有长度至多为\(3\)的模式内部零点的存在性。