A survey is presented of some methods and results on counting words that satisfy various restrictions on subwords (i.e., blocks of consecutive symbols). Various applications to comma-free codes, games, pattern matching, and other subjects are indicated. The emphasis is on the unified treatment of those topics through the use of generating functions. 1.

A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a non-negative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is order-preserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1

Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variable-length codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is constant and need not be generated for every probability distribution. These codes can be used as alternatives to Huffman codes when the optimal compression of the latter is not required, and simplicity, faster processing and robustness are preferred. The codes are compared on several "real-life" examples. 1. Motivation and Introduction Let A = fA 1 ; A 2 ; \Delta \Delta \Delta ; An g be a finite set of elements, called cleartext elements, to be encoded by a static uniquely decipherable (UD) code. For notational ease, we use the term `code' as abbreviation for `set of codewords'; the corresponding encoding and decoding algorithms are always either given or clear from the context. A code i...