Abstract: A permutation set (M, I′) consisting of a setM and a set г of permutations ofM, is calledsymmetric, if for any two permutationsα, β ∈ г the existence of anx ∈ M with α (x) ≠β(x) andα−1β (x) = β−1α(x) impliesα−1β = β−1α, andsharply 3-transitive, if for any two triples (x1,x2,x3), (y1,y2,y3)∈ M3 with|{x1,x2,x3}| = |{y1,y2,y3}| = 3 there is exactly one permutationγ ∈ г withγ(x1) =y1,γ(x2) =y2,γ(x3) =y3. The following theorem will be proved.

Abstract: An equidistant permutation array (E.P.A.)A(r, λ v) is av × r array in which every row is a permutation of the integers 1, 2, ⋯,r such that any two distinct rows have precisely λ columns in common. In this paper we introduce the concept of orthogonality for E.P.A.s. A special case of this is the well known idea of a set of pairwise orthogonal latin squares. We show that a set of these arrays is equivalent to a particular type of resolvable (r, λ)-design. It is also shown that the cardinality of such a set is bounded byr − λ with the upper bound being obtained only ifλ = 0. A brief survey of related orthogonal systems is included. In particular, sets of pairwise orthogonal symmetric latin squares, sets of orthogonal Steiner systems and sets of orthogonal skeins.

Abstract: This chapter describes an algorithm for generating random permutation of n letters. The random permutation is generated by a sequence of random interchanges. First any one of the n letters 1, …, n for a 1 is chosen. Then, any one of the remaining letters for a 2 is chosen and so on. The construction of a permutation, thus, involves n choices, with respective probabilities and the probability of a given permutation chosen being 1/ n ! The FORTRAN program of algorithm RANPER contains a LOGICAL parameter SETUP. If it is set. FALSE., the subprogram will not setup the array A with 1, …, n but, instead, will operate on whatever data the user has supplied. The chapter presents the sample output for the subroutine. In the analysis presented in the chapter, for each n = 3,……,8, a set of 50 random permutations of n letters was chosen and the number of cycles of each of these 50 permutations was found.

Abstract: Let G be a doubly transitive permutation group on a finite set fi, and let K" be a normal subgroup of the stabilizer Ga of a point a in £2. If the action of Ga on the set of orbits of K" in £2—{a} is 2-primitive with kernel K" it is shown that either G is a normal extension of PSL(3, q) or K" n Gy is a strongly closed subgroup of G ay in G a, where y e £2—{a}. If in addition the action of Go on the set of orbits of K" is assumed to be 3-transitive, extra information is obtained using permutation theoreti c and centralizer ring methods. In the case where K" has three orbits in £2—{a} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters ofG.

Abstract: Abstract A branch-and-bound algorithm is presented for the permutation flow-shop problem in which the objective is to minimise the maximum completion time. A branching procedure is used in which jobs both at the beginning and at the end of the schedule have been fixed. Dominance rules are included in the algorithm. Also, during the initial stages of the algorithm, upper bounds are computed at certain nodes of the search tree. Computational results indicate that the proposed algorithm is superior to previously published algorithms.

Abstract: A better algorithm is presented for the generation of an arbitrary permutation in a model of magnetic bubble memories that have been investigated previously.

Abstract: Two permutations on n elements are at (Hamming) distance μ if they disagree in exactly μ places. An equidistant permutation array is a collection of permutations on n elements, every pair of which is at distance μ. A permutation graph G(n,μ) is a graph with vertex set comprising all permutations on n elements, and edges between each pair of permutations at distance μ. These graphs enable the relevant permutation structure to be visualised; in particular, the cliques correspond to maximal equidistant permutation arrays. We obtain various structural theorems for these graphs, and conjecture several properties for their cliques.

Abstract: Reflections on the legitimate deck problem.- Some extremal problems on families of graphs and related problems.- Integral properties of combinatorial matrices.- A class of three-designs.- Isomorphic factorisations III: Complete multipartite graphs.- Biplanes and semi-biplanes.- Near-self-complementary designs and a method of mixed sums.- Recent progress and unsolved problems in dominance theory.- On the linear independence of sets of 2q columns of certain (1, ?1) matrices with a group structure, and its connection with finite geometries.- The doehlert-klee problem: Part I, statistical background.- On the cayley index of a group.- A survey of extremal (r,?)-systems and certain applications.- On the enumeration of certain graceful graphs.- Fixing subgraphs of Km,n.- Hadamard equivalence.- A note on equidistant permutation arrays.- The combinatorics of algebraic graph theory in theoretical physics.- Graphs, groups and polytopes.- Decompositions of complete symmetric digraphs into the four oriented quadrilaterals.- Brick packing.- Colour symmetry in crystallographic space groups.- Generation of a frequency square orthogonal to a 10x10 latin square.- Factorization in the monoid of languages.- On graphs as unions of eulerian graphs.- The analysis of colour symmetry.- Computing automorphisms and canonical labellings of graphs.- On a result of bose and shrikhande.- Further results on a problem in the design of electrical circuits.- Transversals and finite topologies.- Asymptotic number of self-converse oriented graphs.- Some correspondences involving the schroder numbers and relations.- A computer listing of hadamard matrices.- A class of codes generated by circulant weighing matrices.- An application of maximum-minimum distance circuits on hypercubes.- Decompositions of graphs and hypergraphs.- Some extremal problems in combinatorial geometry.- Distance-regular graphs.- A note on baxter's generalization of the temperley-lieb operators.- Autocorrelation of (+1,?1) sequences.- Triangles in labelled cubic graphs.- Problems.

Abstract: Branch-and-bound methods are commonly used to find a permutation schedule that minimizes maximum completion time in an m-machine flow-shop. In this paper we describe a classification scheme for lower bounds that generates most previously known bounds and leads to a number of promising new ones as well. After a discussion of dominance relations within this scheme and of the implementation of each bound, we report on computational experience that indicates the superiority of one of the new bounds.