The Cycle Construction

We give a direct generating function construction for cycles of combinatorial structures. Let A be a class of combinatorial structures, with A(z) its corresponding ordinary generating function: A(z) = P 2A z jj. We use corresponding letters for classes and generating functions. Consider the class C whose elements are cycles of elements of A. The following result is classical 6], 1]: C(z) = X k1 (k) k log 1 1 ? A(z k) ; (0) where (k) is the Euler totient function. This result is proved by Read 6] using PP olya theory 5] and a classical computation of the Zyklenzeichner of the cyclic group. De Bruijn and Klarner 1] have another derivation, which amounts to the Lyndon factorization of free monoids 4, p. 64]. Our purpose in this note is to show that equality (0) follows directly from basic principles of combinatorial analysis 3], using elementary concepts of combinatorics on words from Lothaire 4]. Principle 1. Every non-empty word over A has a unique root which is a primitive word. For instance with ; 2 A, word decomposes into jj and its root is the primitive (also called aperiodic) word. Let S = A + be the set of non-empty words formed with elements of A, and PS the set of primitive words. From Principle 1, we havez S(z; u) uA(z) 1 ? uA(z) = X k1 PS(z k ; u k) : (1a) From Moebius inversion applied to (1a), we get an explicit form for PS(z; u): PS(z; u) = X k1 (k)S(z k ; u k) = X k1 (k) u k A(z k) 1 ? u k A(z k) : (1b) Principle 2. Every primitive k-cycle has k distinct primitive word representations. A cycle is said to be primitive ii any associated word is primitive. We use the notation. . .] to denote a cycle. Then, for instance, the 5-cycle ababb] = babba] =. .. = babab] is primitive, while the 6-cycle abbabb] is not. We let PC denote the class of primitive cycles. Principle 2 permits to express the bivariate generating function PC(z; u) via the transformation u k 7 ! u k =k applied to PS(z; u): PC(z; u) = Z u 0 PS(z; t) dt t : (2a) Integrating with respect to u, we derive PC(z; u) = X k1 (k) k log 1 1 ? u k A(z k) : (2b) y …