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.     

On permutation labeling

We determine all permutation graphs of order ?9. We prove that every bipartite graph of order ?50 is a permutation graph. We convert the conjecture stating that “every tree is a permutation graph” to be “every bipartite graph is a permutation graph”.     

A Note on Point Stabilizers in Sharp Permutation Groups of Type {0, k}

We study sharp permutation groups of type {0, k} and observe that, once the isomorphism type of a point stabilizer is fixed, there are only finitely many possibilities for such a permutation group. We then show that a sharp permutation group of type {0, k} in which a point stabilizer is isomorphic to the alternating group on 5 letters must be a geometric group. There is, up to permutation isomorphism, one such permutation group.     

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.     

On permutation graphs

We give an upper bound of the number of edges of a permutation graph. We introduce some necessary conditions for a graph to be a permutation graph, and we discuss the independence of these necessary conditions. We show that they are altogether not sufficient for a graph to be a permutation graph.     

对换改变排列奇偶性的一个定量证明

先对排列逆序及逆序数的概念进行拓展,然后再给出关于对换改变排列奇偶性这个定理的一个定量证明.     

Permutation binomials and their groups

This paper is devoted to studying the properties of permutation binomials over finite fields and the possibility to use permutation binomials as encryption functions. We present an algorithm for enumeration of permutation binomials. Using this algorithm, all permutation binomials for finite fields up to order 15000 were generated. Using this data, we investigate the groups generated by the permutation binomials and discover that over some finite fields \mathbb F_q {{\mathbb F}_q} , every bijective function on [1..q − 1] can be represented as a composition of binomials. We study the problem of generating permutation binomials over large prime fields. We also prove that a generalization of RSA using permutation binomials is not secure. Bibliography: 9 titles.     

On Some Properties of Permutation Tableaux

We consider the relations between various permutation statistics and properties of permutation tableaux. We answer some of the open problems of Steingrímsson and Williams [8], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations. Research supported in part by the NSA Young Investigator Grant H98230-06-1-0037.     

A direct bijection between descending plane partitions with no special parts and permutation matrices

We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.     

Exclusion processes with multiple interactions

We introduce the mathematical theory of the particle systems that interact via permutations, where transition rates are assigned not to the jumps from a site to a site, but to the permutations themselves. These permutation processes can be viewed as the natural generalization of symmetric exclusion processes, where particles interact via transpositions. We develop a number of innovative coupling techniques for the permutation processes and establish the needed conditions for them to apply. We use duality, couplings and other tools to explore the stationary distributions of the permutation processes with translation invariant rates.     

Pattern Avoidance in Ordered Set Partitions

In this paper we consider the enumeration of ordered set partitions avoiding a permutation pattern of length 2 or 3. We provide an exact enumeration for avoiding the permutation 12. We also give exact enumeration for ordered partitions with 3 blocks and ordered partitions with n?1 blocks avoiding a permutation of length 3. We use enumeration schemes to recursively enumerate 123-avoiding ordered partitions with any block sizes. Finally, we give some asymptotic results for the growth rates of the number of ordered set partitions avoiding a single pattern; including a Stanley-Wilf type result that exhibits existence of such growth rates.     

Non-nudgable subgroups of permutations

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.     

Asymptotic normality and combinatorial aspects of the prefix exchange distance distribution

The prefix exchange distance of a permutation is the minimum number of exchanges involving the leftmost element that sorts the permutation. We give new combinatorial proofs of known results on the distribution of the prefix exchange distance for a random uniform permutation. We also obtain expressions for the mean and the variance of this distribution, and finally, we show that the normalised prefix exchange distribution converges in distribution to the standard normal distribution.     

Multi-avoidance of generalised patterns

Recently, Babson and Steingrímsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3-patterns without internal dashes, that is, where the pattern corresponds to a contiguous subword in a permutation.     

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.     

Split Permutation Graphs

The class of split permutation graphs is the intersection of two important classes, the split graphs and permutation graphs. It also contains an important subclass, the threshold graphs. The class of threshold graphs enjoys many nice properties. In particular, these graphs have bounded clique-width and they are well-quasi-ordered by the induced subgraph relation. It is known that neither of these two properties is extendable to split graphs or to permutation graphs. In the present paper, we study the question of extendability of these two properties to split permutation graphs. We answer this question negatively with respect to both properties. Moreover, we conjecture that with respect to both of them the split permutation graphs constitute a critical class.     

Check Character Systems using Quasigroups: I

We consider check character systems with one check character over an arbitrary quasigroup under different check equations which have one permutation. The aim of using such a system is to discover transmission errors. These systems always detect all single errors (that is errors only one component of a code word). We establish necessary and sufficient conditions for a general quasigroup check character system over this quasigroup to be able to detect all transpositions. jump transpositions, twin errors and jump twin errors. We also study capabilities of check character systems over loops, in particular, over Moufang loops and groups when the permutation in the check equations is the identify permutation.Communicated by: P. Wild     

SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.     

Permutation presentations of modules over finite groups

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.     

Diagonally neighbour transitive codes and frequency permutation arrays

Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication, while at the same time maintaining a constant power output. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family.