Staircase words and chebyshev polynomials

Abstract

A word σ = σx · · · σnover the alphabet [k] = {1, 2,..., k} is said to be a staircase if there are no two adjacent letters with difference greater than 1. A word σ is said to be staircase-cyclic if it is a staircase word and in addition satisfies |σn - σ| ≤ 1. We find the explicit generating functions for the number of staircase words and staircase-cyclic words in [k]n, in terms of Chebyshev polynomials of the second kind. Additionally, we find explicit formulæ for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate staircase necklaces, which are staircase-cyclic words that are not equivalent up to rotation.