In particular, we give various necessary conditions for w to be in C(u). We also characterize C(u) when u has few letters, when it has a single repeated entry, or when it is a certain type of decreasing sequence.
We consider cn,m(u), the number of w ∈ C(u) of length n with max w ≤ m.
We prove that for |u|=1 the value of this function depends only on the relative sizes of u and m and not on their actual values. And for various u we use Stanley's theory of poset partitions to show that, for fixed n, cn,m(u) is a polynomial in m with certain degree and leading coefficient. We end with various conjectures and directions for further research.
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