Several classes of complete permutation polynomials over finite fields of even characteristic

In this paper, we find three classes of complete permutation polynomials over finite fields of even characteristic. The first class of quadrinomials is complete in the sense of addition. The second and third classes of binomials and trinomials are complete in multiplication. Moreover, a result related to the complete property in multiplication of a special class of polynomials is also given.     

Linear optimization over permutation groups

For a permutation group given by a set of generators, the problem of finding “special” group members is NP-hard in many cases, e.g., this is true for the problem of finding a permutation with a minimum number of fixed points or a permutation with a minimal Hamming distance from a given permutation. Many of these problems can be modeled as linear optimization problems over permutation groups. We develop a polyhedral approach to this general problem and derive an exact and practically fast algorithm based on the branch & cut-technique.     

New classes of permutation binomials and permutation trinomials over finite fields

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication theory and so on. Permutation binomials and permutation trinomials attract people's interest due to their simple algebraic forms and additional extraordinary properties. In this paper, we find a new result about permutation binomials and construct several new classes of permutation trinomials. Some of them are generalizations of known ones.     

Coloring permutation graphs in parallel

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having specific key properties. We propose an efficient parallel algorithm which colors an n-node permutation graph in O(log² n) time using O(n²/log n) processors on the CREW PRAM model. Specifically, given a permutation π we construct a tree T^*[π], which we call coloring-permutation tree, using certain combinatorial properties of π. We show that the problem of coloring a permutation graph is equivalent to finding vertex levels in the coloring-permutation tree.     

Short proofs for cut-and-paste sorting of permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of [n] using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of [n] can be transformed to the identity in at most ⌊2n/3⌋ such moves and that some permutations require at least ⌊n/2⌋ moves.     

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ?4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.     

A negative answer to Bracken–Tan–Tan's problem on differentially 4-uniform permutations over F2n

Permutations with low differential uniformity are widely used in cipher design. Recently, Bracken, Tan and Tan (2012) [5] presented a method to construct differentially 4-uniform permutations by changing certain conditions of known APN functions. They guessed that only two classes of existing quadratic APN functions have this property. They succeeded in proving one class and left the other one as an open problem. In this paper, with the help of a computer, those polynomials are proved to be differentially 4-uniform but may not be permutation polynomials, which give a negative answer to this problem.     

The harmonious coloring problem is NP-complete for interval and permutation graphs

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.     

Asymptotic bounds for permutations containing many different patterns

We say that a permutation σ∈Sn contains a permutation π∈Sk as a pattern if some subsequence of σ has the same order relations among its entries as π. We improve on results of Wilf, Coleman, and Eriksson et al. that bound the asymptotic behavior of pat(n), the maximum number of distinct patterns of any length contained in a single permutation of length n. We prove that by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of k-superpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that Lk, the length of the shortest k-superpattern, is at most . This may lend evidence to a conjecture of Eriksson et al. that .     

Permutation tableaux and permutation patterns

In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the -diagrams of Alex Postnikov [A. Postnikov, Webs in totally positive Grassmann cells, in preparation; L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative sizes of pairs of letters in a permutation and their places. We call these statistics weak excedance statistics because of their close relation to weak excedances. The second bijection carries tableaux statistics (via the weak excedance statistics) to statistics based on generalized permutation patterns. We then give enumerative applications of these bijections. One nice consequence of these results is that the polynomial enumerating permutation tableaux according to their content generalizes both Carlitz' q-analog of the Eulerian numbers [L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350] and the more recent q-analog of the Eulerian numbers found in [L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. We conclude our paper with a list of open problems, as well as remarks on progress on these problems which has been made by A. Burstein, S. Corteel, N. Eriksen, A. Reifegerste, and X. Viennot.     

A note on the total completion time problem in a permutation flowshop with a learning effect

The concept of learning process plays a key role in production environments. However, it is relatively unexplored in the flowshop setting. In this short note, we consider a permutation flowshop scheduling problem with a learning effect where the objective is to minimize the sum of completion times or flowtime. A dominance rule and several lower bounds are established to speed up the search for the optimal solution. In addition, the performances of several well-known heuristics are evaluated when the learning effect is present.     

A scatter search algorithm for the distributed permutation flowshop scheduling problem

The distributed permutation flowshop problem has been recently proposed as a generalization of the regular flowshop setting where more than one factory is available to process jobs. Distributed manufacturing is a common situation for large enterprises that compete in a globalized market. The problem has two dimensions: assigning jobs to factories and scheduling the jobs assigned to each factory. Despite being recently introduced, this interesting scheduling problem has attracted attention and several heuristic and metaheuristic methods have been proposed in the literature. In this paper we present a scatter search (SS) method for this problem to optimize makespan. SS has seldom been explored for flowshop settings. In the proposed algorithm we employ some advanced techniques like a reference set made up of complete and partial solutions along with other features like restarts and local search. A comprehensive computational campaign including 10 existing algorithms, together with statistical analyses, shows that the proposed scatter search algorithm produces better results than existing algorithms by a significant margin. Moreover all 720 known best solutions for this problem are improved.     

An exact parallel method for a bi-objective permutation flowshop problem

In this paper, we propose a parallel exact method to solve bi-objective combinatorial optimization problems. This method has been inspired by the two-phase method which is a very general scheme to optimally solve bi-objective combinatorial optimization problems. Here, we first show that applying such a method to a particular problem allows improvements. Secondly, we propose a parallel model to speed up the search. Experiments have been carried out on a bi-objective permutation flowshop problem for which we also propose a new lower bound.     

Constructing permutation polynomials from piecewise permutations

We present a construction of permutation polynomials over finite fields by using some piecewise permutations. Based on a matrix approach and an interpolation approach, several classes of piecewise permutation polynomials are obtained.     

Permutation trinomials over F2m

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over ${\mathbb{F}}_{2^m}$ in Zieve's paper [30]. We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over ${\mathbb{F}}_{2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.     

On inverses of some permutation polynomials over finite fields of characteristic three

By using the piecewise method, Lagrange interpolation formula and Lucas' theorem, we determine explicit expressions of the inverses of a class of reversed Dickson permutation polynomials and some classes of generalized cyclotomic mapping permutation polynomials over finite fields of characteristic three.     

Equivalence of permutation polytopes corresponding to strictly supermodular functions

This paper demonstrates a strong equivalence of all permutation polytopes corresponding to strictly supermodular functions.     

Permutation polytopes and indecomposable elements in permutation groups

Each group G of n×n permutation matrices has a corresponding permutation polytope, P(G):=conv(G)⊂Rn×n. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min{2t,⌊n/2⌋} is a sharp upper bound on the diameter of the graph of P(G). We also show that P(G) achieves its maximal dimension of ²(n−1) precisely when G is 2-transitive. We then extend the results of Pak [I. Pak, Four questions on Birkhoff polytope, Ann. Comb. 4 (1) (2000) 83-90] on mixing times for a random walk on P(G). Our work depends on a new result for permutation groups involving writing permutations as products of indecomposable permutations.