Sequence enumeration

The purpose of this paper is to present a method for treating uniformly a large number of enumeration problems involving sequences over a finite alphabet subject to general restrictions. Many of these problems are classical in origin, including the Mrnage problem (Lucas [16]), the alternating permutation problem (Netto [19], Andr6 [1, 2]), the derangement problem (Montmort [18]), and more recently, the Simon Newcomb problem (Riordan [21]) and the Smirnov problem (Smirnov, Saramanov, Zaharov [23]). Renewed interest in sequence enumeration has been shown in the recent literature. In a given situation, we obtain a regular expression which generates all permissible sequences. The generating function associated with the problem is obtained by a simple transformation of this expression. This process of transformation has a direct combinatorial basis and involves the insertion of indeterminares to record the appropriate combinatorial information. Thus, we consider all sequences in a class and choose those with the required property by extracting the coefficient of the appropriate power of the indeterminate. Additive and multiplicative properties of these generating functions are given, which allow us to induce a linear system of equations for the required function. The solution, obtained by Cramer's Rule, consists of generating functions for elementary objects. These in turn are easily found by combinatorial means. Our concern in this paper is to present a general method, together with a few examples which illustrate the ways in which the method is used. The reference which is cited for each result is, as far as it is possible to discover, a reference to the first solution, usually by other means. Several proofs are omitted. These are accessible in the literature and concern algebraic devices used in the theory.