The shuffle-exchange permutation network (SEPn) is a fixed degree Cayley graph which has been proposed as a basis for massively parallel systems. We propose a routing algorithm with an upper bound of (5/8)n + O(n), where n is the length of the permutation. (This improves on a (9/8)n routing algorithm described earlier [5].) Thus, the diameter of SEPn is at most (5/8) n 2 + O(n). We also show th...

We consider the triangle-free process: Given an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph Kn. Then, traverse the edges of Kn according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving graph unless its addition creates a triangle. We study the evolving graph at around the time where Θ(...

A tower is a polygon consisting of two reflex chains sharing one common endpoint, together with one edge joining the other endpoints of the chains. A linear time algorithm is given to recognize the [vertex] visibility graphs of towers, and these graphs are characterized as bipartite permutation graphs with an added Hamiltonian cycle. Similar results have been obtained independently by Choi, Shi...

In this paper a new interleaver is investigated, namely the permutation polynomial P(x) = (a0x + a1) + a2 over a prime field Fp, and it is shown that it has some favourable properties. The exact value of dispersion of P(x) is given in terms of the parameters a0, a1. Results on the construction and enumeration of permutations P(x) with prescribed dispersion (e.g. close to that of a random interl...

With any permutation g of a set Ω is associated a partition of Ω into the cycles of g. What information do we get about a group G of permutations if we know either the set or the multiset of partitions of Ω, or of partitions of n = |Ω|, which arise as the cycle partitions of its elements? Some partial answers to these questions are given.

Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism group). We prove the fixing numbers for the cycle matroid and bicircular matroid associated with 3-connected graphs are identical. Many of these results have...

Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. This paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of pattern containment. These techniques are used to prove a new characterization of vexillary permutations in terms of their principal dual order ideals in a ...