Given an alphabet S, we consider the size of the subsets of the full
sequence space SZ determined by the additional restriction that
xi=xi+f(n),i∈Z,n∈N. Here f is a
positive, strictly increasing function. We review an other, graph theoretic,
formulation and then the known results covering various combinations of f and
the alphabet size. In the second part of the paper we turn to the fine
structure of the allowed sequences in the particular case where f is a
suitable polynomial. The generation of sequences leads naturally to consider
the problem of their maximal length, which turns out highly random
asymptotically in the alphabet size.Comment: 18 pages, 3 figures. Replaces earlier version, submission 1204.3439,
major updat