Longest alternating subsequences of k-ary words

Recently, Stanley [Longest alternating subsequences of permutations, preprint, arXiv/0511419v1] studied the length of the longest alternating subsequence of a permutation in the symmetric group, where a sequence a, b, c, d,... is alternating if a > b < c > d < .... In this paper, we extend this result to the case of k-ary words. More precisely, we find an explicit formula for the generating function of the number of k-ary words of length n according to the length of the longest alternating subsequence. (c) 2007 Elsevier B.V. All rights reserved.