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A word $sigma=sigma_1...sigma_n$ 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 $sigma$ is said to be {em smooth cyclic} if it is a smooth word and in addition satisfies $|sigma_n-sigma_1|le 1$. 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.
We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated weights as well as an arbitrary `background weight. Our CT theorem, like Viennots lattice
In the present paper we find a simple algorithm for counting Jacobian group of the generalized Petersen graph GP(n,k). Also, we obtain a closed formula for the number of spanning trees of this graph in terms of Chebyshev polynomials.
We show how polynomial mappings of degree k from a union of disjoint intervals onto [-1,1] generate a countable number of special cases of a certain generalization of the Chebyshev Polynomials. We also derive a new expression for these generalized Ch
Given $n$ polynomials $p_1, dots, p_n$ of degree at most $n$ with $|p_i|_infty le 1$ for $i in [n]$, we show there exist signs $x_1, dots, x_n in {-1,1}$ so that [Big|sum_{i=1}^n x_i p_iBig|_infty < 30sqrt{n}, ] where $|p|_infty := sup_{|x| le 1} |p(
Orbit functions of a simple Lie group/Lie algebra L consist of exponential functions summed up over the Weyl group of L. They are labeled by the highest weights of irreducible finite dimensional representations of L. They are of three types: C-, S- a