Enumerating Row Arrangements of Three Species

Abstract

Introduction A classic problem in beginning finite or discrete mathematics courses reads as follows: A photographer wants to arrange N men and N women in a line so that men and women alternate. In how many ways can this be done? This problem nicely illustrates the use of factorials and has a simple solution, though students often neglect the factor of two in the answer 2(N!)2. This omission can be instructive, as it leads naturally to generalizations of the problem: How does the answer change if there are N men and N 1 women? What if men outnumber women by 2 or more? What if a sexist photographer insists that the lineup start with a man? These variations are all easily dealt with, and illustrate some of the possible subtleties encountered in counting problems. Another variation of the problem is not, however, so easily handled. The setting (or sitting) must change slightly: A pet photographer must pose C cats, D dogs, and E emus in a line, with no two animals of the same species adjacent. In how many ways can this be done? Skirting the issue of whether emus can in fact be considered pets, we quickly discover many interesting features in this variation of the problem. For example, C, D, and E can differ by more than 1 and still admit nonzero solutions. We also discover that the new problem is quite a bit harder than the original. Dating back to 1966 ([6]), it is often called Smirnov's problem and it has practical applications in areas such as queuing, transportation flow, sequential analysis, and multivariate order statistics (see, e.g., [1, 5]). The problem has been solved in a variety of ways (some of which we mention at the end of this note) but, unlike those methods, what is presented here is entirely elementary and could be discussed in a first course in discrete mathematics or combinatorics.