光滑字与切比雪夫多项式
Smooth words and Chebyshev polynomials
摘要：
字母表$[k]=\{1,2,...,k\}$上的单词$σ=σ_1...σ_n$被称为{ \em smooth}，如果没有两个相邻字母的差大于1。一个词$σ$被称为{ \em smooth cyclic}，如果它是一个平滑词，并且满足$|σ_n-σ_1|\le 1$。我们找到了$[k]^n$中光滑字数和循环光滑字数的显式生成函数，用{ \it第二类切比雪夫多项式}表示.此外，我们发现显式公式的数字本身，作为三角和。这些导致直接的渐近推论。我们还列举了光滑项链，这是循环光滑的话，不等于旋转。
英文摘要：
A word $σ=σ_1...σ_n$ over the alphabet $[k]=\{1,2,...,k\}$ is said to be {\em smooth} if there are no two adjacent letters with difference greater than 1. A word $σ$ is said to be {\em smooth cyclic} if it is a smooth word and in addition satisfies $|σ_n-σ_1|\le 1$. We find the explicit generating functions for the number of smooth words and cyclic smooth words in $[k]^n$, in terms of {\it Chebyshev polynomials of the second kind}. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate smooth necklaces, which are cyclic smooth words that are not equivalent up to rotation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
