Recently, Bouvel and Pergola initiated the study of a special class of permutations, minimal permutations with a given number of descents, which arise from the whole genome duplication-random loss model of genome rearrangement. In this paper, we show that the number of minimal permutations of length 2d− 1 with d descents is given by 2d−3(d− 1)cd, where cd is the d-th Catalan number. We also der...

*Correspondence: [email protected] 1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N. W., Calgary, AB T2N 1N4, Canada Full list of author information is available at the end of the article Abstract In this paper, we report an error in the paper of the first author in Advances in Difference Equations, 2009, article 104310, present the revised versio...

In this work we count the number of involutory, unimodal, and alternating elements of the group of signed permutations Bn, and the group of even-signed permutations Dn. Recurrence relations, generating functions, and explicit formulas of the enumerating sequences are given. © 2006 Elsevier B.V. All rights reserved. MSC: primary: 05A15; secondary: 05A19; 05A05

Let [T (n, k)]n,k>0 be a triangle of positive numbers satisfying the three-term recurrence relation T (n, k) = (a1n + a2k + a3)T (n− 1, k) + (b1n + b2k + b3)T (n− 1, k − 1). In this paper, we give a new sufficient condition for linear transformations

Introduction: Early recurrence of hepatocellular carcinoma (HCC) is a major risk factor affecting survival even after hepatectomy. Many clinical, biochemical parameters and pathological grading like fibrosis 1 index have been used for risk stratifying HCC. However not many studies have combined all of them. It is therefore important to risk stratify HCC especially with newer PE...

In this paper, we investigate the global behavior of the difference equation xn+1 = αxn−1 β + γ t ∑