Structured Matrices, Continued Fractions, and Root Localization of Polynomials

Restricted Permutations, Continued Fractions, and Chebyshev Polynomials

Let fr n(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let Fr(x; k) and F (x, y; k) be the generating functions defined by Fr(x; k) = P n>0 f r n(k)x n and F (x, y; k) = P r>0 Fr(x; k)y r. We find an explicit expression for F (x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 6 r 6 k via Cheb...

2d Continued Fractions and Positive Matrices

The driving force of this paper is a local symmetry in lattices. The goal is two theorems: a partial converse to the Perron-Frobenius theorem in dimension 3 and a characterization of conjugacy in Sl(Z). In the process we develop a geometric approach to higher dimension continued fractions, HDCF. HDCF is an active area with a long history: see for example Lagarias, [L],[Br]. The algorithm: Let Z...

0 Restricted Permutations , Continued Fractions , and Chebyshev Polynomials

Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials

We say that a permutation is a Motzkin permutation if it avoids 132 and there do not exist a <b such that a < b < b+1. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribut...

Multivariate Polynomials, Duality, and Structured Matrices

We rst review the basic properties of the well known classes of Toeplitz Hankel Vandermonde and other related structured matrices and re examine their correlation to operations with univariate polynomials Then we de ne some natural extensions of such classes of matrices based on their correlation to multivariate polynomials We describe the correlation in terms of the associated operators of mul...