In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, we make use of the kernel method and obtain an explicit formula (in terms of Eulerian polynomials) for the distribution on Sn in the flattened ...

Abstract Suppose a large and dense point cloud of an object with complex geometry is available that can be approximated by smooth univariate function. In general, for such clouds the “best” approximation using method least squares usually hard or sometimes even impossible to compute. most cases, however, “near-best” just as good “best”, but much easier faster calculate. Therefore, fast approach...

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre poly...

This paper investigates certain Jacobi polynomials that involve one parameter and generalize the well-known orthogonal called Chebyshev of third-kind. Some new formulas are developed for these polynomials. We will show some previous results in literature can be considered special ones our derived formulas. The derivatives moments derived. Hence, two important explicitly give terms their origina...

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of p(A) over all monic polynomials p(z) of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well known propertie...

We study generating functions for the number of involutions of length n avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation τ of length k. In several interesting cases these generating functions depend only on k and can be expressed via Chebyshev polynomials of the second kind. In particular, we show that involutions of length n avoiding ...

A partition π of the set [n] = {1, 2, . . . , n} is a collection {B1, B2, . . . , Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. In this paper, we find an explicit formula for the generating function for the number of partitions of [n] with exactly k blocks according to the number of peaks (valleys) in terms of Chebyshev polynomials of the second kind. Furthermo...

given function f imperative to have certain smoothness or continuity properties. Particular attention is paid to error estimate of the developed AQS, where it shows the acquired AQS scheme is obtained in the class of functions C[−1, 1] which converges to the exact very fast by increasing the knot points. The first and second kind of Chebyshev polynomials are used in the conjecture. Several nume...