Longest run of equal parts in a random integer composition

has L = 5. A composition with L = 1 is known as a Carlitz composition. The characteristics of Carlitz compositions and their generating function C〈1〉(z) (see Proposition 1) are studied in great detail in [4, 5]. The solution to the longest run problem can be broken down into four main sections. In the first section, we find a family of generating functions for integer compositions that keeps track of the longest run of equal parts. In the second section, we analyze the generating functions using singularity analysis to find an asymptotic estimate of the number of compositions of size n with no run of length k. In the third, we use that estimate to describe the probability distribution of the random variable L, and in the fourth, we calculate the mean and variance of the distribution. The analysis here has some similarities to the analytic treatment of compositions in [1, 4, 5], and the methods and notation used in this note are detailed in the book Analytic Combinatorics by Flajolet and Sedgewick [3]. This note was motivated by a question of Wilf, posed at the Analysis of Algorithms 09 Conference (Frejus, June 2009); see [6]. The author would like to thank Herbert Wilf for suggesting this problem and Philippe Flajolet for his direction and support throughout this project. ∗This work was done during a summer internship at Algorithms Project, INRIA-Rocquencourt, F78153 Le Chesnay, France, May-July, 2009, under the direction of Philippe Flajolet. Author’s permanent email address is [email protected].