Smooth compositions and smooth words

A composition of a positive integer n, ? = ?1?2 ?N, where ?1 ?2 ?N = n, is said to be smooth if it contains no pair of adjacent letters with difference greater than 1. A smooth composition ? is called cyclic if in addition it satisfies |?1 ? ?N | ? 1. In this paper we study the problem of enumerating the smooth compositions of n with parts in a set. We obtain generating functions for the numbers of smooth compositions and smooth cyclic compositions of n with parts in the set {1, . . . , k}. We also derive asymptotic estimates for the numbers of the compositions via singularity analysis. Finally, by viewing compositions as a restricted class of words, we deduce several results on smooth words, including previously known ones.