In 1966, Claude Berge proposed the following sorting problem. Given a string of n alternating white and black pegs, rearrange the pegs into a string consisting of ⌈n 2 ⌉ white pegs followed immediately by ⌊n 2 ⌋ black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Berge’s original question was generalized by considering the same sorting problem usin...

In a previous paper [3], the author gave an explicit description of the algebraic closure of the power series field over a field of characteristic p > 0, in terms of certain “generalized power series”. The purpose of the present paper is to extend this work to mixed characteristic. Specifically, we give an analogous description of the algebraic closure of the Witt ring W (K) of an algebraically...

In this paper, we present an algorithmic approach to nd the stationary probability distribution of M/G/1 type Markov chains which arise frequently in performance analysis of computer and communication networks. The approach uniies nite-and innnite-level Markov chains of this type through a generalized state-space representation for the probability generating function of the stationary solution....

Abstract. A k-generalized Dyck path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (k, 0) for a given integer k ≥ 0, up-steps (1, 1), and down-steps (1,−1), which never passes below the x-axis. The present paper studies three kinds of statistics on k-generalized Dyck paths: ”number of u-segments”, ”number of internal u-segme...

Various generalizations of the Fibonacci numbers have been proposed, studied and applied over the years (see [5] for a brief list). Probably the best known are the k-generalized Fibonacci numbers F (k) n (also known as the k-fold Fibonacci, k-th order Fibonacci, k-Fibonacci or polynacci numbers), satisfying F (k) n = F (k) n−1 + F (k) n−2 + . . .+ F (k) n−k , n ≥ k , F (k) n = 0 , 0 ≤ n ≤ k − 2...

In the study of permutations, generalized patterns extend classical patterns by allowing the requirement that certain adjacent integers in a pattern must be adjacent in the permutation. For any generalized pattern π∗ 0 of length k with 1 ≤ b ≤ k blocks, we prove that for all μ > 0, there exists 0 < c = c(k, μ) < 1 so that whenever n ≥ n0(k, μ, c), all but cnn! many π ∈ Sn admit (1± μ) 1 k! (n b...

Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized k-connectivity of undirected graphs to directed graphs (we call it stro...

We study the existence of generalized transition fronts for a bistable reaction diffusion equation, ut − ∆u = f(u), in a heterogeneous medium, Ω = RN\K where K is a compact set of RN , with Neumann boundary condition and t ∈ R. In the paper, Bistable traveling waves around an obstacle (2009), H. Berestycki, F. Hamel and H. Matano prove the existence of a generalized transition front when K is s...

We propose a new family of coherence monotones, named the generalized coherence concurrence (or coherence k-concurrence), which is an analogous concept to the generalized entanglement concurrence. The coherence k-concurrence of a state is nonzero if and only if the coherence number (a recently introduced discrete coherence monotone) of the state is not smaller than k, and a state can be convert...

In this paper, we study the properties of the generalized harmonic numbersHn,k,r(α, β). In particular, by means of the method of coefficients, generating functions and Riordan arrays, we establish some identities involving the numbers Hn,k,r(α, β), Cauchy numbers, generalized Stirling numbers, Genocchi numbers and higher order Bernoulli numbers. Furthermore, we obtain the asymptotic values of s...