RESTRICTED 132-AVOIDING k-ARY WORDS, CHEBYSHEV POLYNOMIALS, AND CONTINUED FRACTIONS

We study generating functions for the number of n-long k-ary words that avoid both 132 and an arbitrary `-ary pattern. In several interesting cases the generating function depends only on ` and is expressed via Chebyshev polynomials of the second kind and continued fractions. 1. Extended abstract 1.1. Permutations. Let Sn denote the set of permutations of [n] = {1, 2, . . . , n}, written in one-line notation, and suppose π ∈ Sn. We write πi to denote the ith element of π, for i = 1, 2, . . . , n. Let π ∈ Sn and τ ∈ Sk be two permutations. We say that π contains τ if there exists a subsequence πi1 , . . . , πik , where 1 ≤ i1 < i2 < · · · < ik ≤ n, such that it is order-isomorphic to τ ; in such a context τ is usually called a pattern. We say that π avoids τ , or is τ -avoiding , if such a subsequence does not exist. The set of all τ -avoiding permutations in Sn is denoted by Sn(τ). For an arbitrary finite collection of patterns T , we say that π avoids T if π avoids any τ ∈ T ; the corresponding subset of Sn is denoted by Sn(T ). For example, the permutation 562314 avoids 132, but it has 634 as a subsequence so it does not avoid 312. While the case of permutations avoiding a single pattern has attracted much attention, the case of multiple pattern avoidance remains less investigated. In particular, it is natural, as the next step, to consider permutations avoiding pairs of patterns τ , τ. Several recent papers [5, 8, 7, 9, 10, 11] deal with the case τ ∈ S3, τ ∈ Sk for various pairs τ , τ. The tools involved in these papers include continued fractions, Chebyshev polynomials of the second kind, and Dyck words. For example, Chow and West [5] have show that