The paper offers an overview over selected results in the literature on partially ordered patterns (POPs) in permutations, words and compositions. The POPs give rise in connection with co-unimodal patterns, peaks and valleys in permutations, Horse permutations, Catalan, Narayana, and Pell numbers, bi-colored set partitions, and other combinatorial objects.

In this paper, we firstly present a general expression for the entries of the th r   N r power of certain -square n are complex tridiagonal matrix, in terms of the Chebyshev polynomials of the first kind. Secondly, we obtain two complex factorizations for Fibonacci and Pell numbers. We also give some Maple 13 procedures in order to verify our calculations.

In this study, we define a n × n tridiagonal matrix which have elements of (s, t)-Pell numbers and then investigate the determinantal properties.

Xanthomonas oryzae pv.oryzae (Xoo) causes the serious bacterial blight disease of rice. Xoo secretes a repertoire of plant cell wall degrading enzymes (CWDEs) like cellulases, xylanases, esterases etc., which act on various components of the rice cell wall. The major cellulases and xylanases secreted by Xoo have been identified and their role in virulence has been determined. In this study, we ...

Let ck = P 2 2k + 1, where Pk denotes the k th Pell number. It is proved that for all positive integers k all solutions of the system of simultaneous Pellian equations z − ckx = ck − 1, 2z − cky = ck − 2 are given by (x, y, z) = (0,±1,±P2k). This result implies that there does not exist positive integers d > c > 2 such that the product of any two distinct elements of the set {1, 2, c, d} dimini...

The plant-pathogenic bacterium Dickeya dadantii produces several pectinolytic enzymes that play a major role in the soft-rot disease. Eight characterized endopectate lyases are secreted in the extracellular medium by the type II secretion system, Out. They cleave internal glycosidic bonds of pectin, leading to plant tissue maceration. The D. dadantii pectate lyases belong to different families,...

1 On the so–called Pell–Fermat equation 2 1.1 Examples of simple continued fractions . . . . . . . . . . . . . 2 1.2 Existence of integer solutions . . . . . . . . . . . . . . . . . . 5 1.3 All integer solutions . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 On the group of units of Z[ √ D] . . . . . . . . . . . . . . . . . 8 1.5 Connection with rational approximation . . . . . . . . . . ....

We provide a new elementary proof of a criterion, given in earlier work, for the solvability of the Pell equation x2 −Dy2 = −1 where D is any positive nonsquare integer. Mathematics Subject Classification: Primary 11D09; 11A55; Secondary 11R11; 11R29