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Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n = mλ and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency λ, such that the Ham-ming distance between any distinct x, y ∈ T is at least d. Such… (More)

A simple permutation is one that does not map a nontrivial interval onto an interval. It was recently proved by Albert and Atkin-son that a permutation class with only finitely simple permutations has an algebraic generating function. We extend this result to enumerate permutations in such a class satisfying additional properties, e.g., the even… (More)

This paper introduces and studies the notion of an equidistant frequency permutation array (EFPA). An EFPA of length n = mλ, distance d and size v is defined to be a v×n array T such that 1) each row is a multipermutation on a multiset of m symbols, each repeated with frequency λ, and 2) the Hamming distance between any two distinct rows of T is precisely… (More)

An element α of the extension E of degree n over the finite field F = GF (q) is called free over F if {α, α q n−1 } is a (normal) basis of E/F. The primitive normal basis theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F , there exists an element α ∈ E such that α is simultaneously primitive (i.e.,… (More)

We prove that every sufficiently long simple permutation contains two long almost disjoint simple subsequences, and then we show how this result has enumerative consequences. For example, it implies that, for any r, the number of permutations with at most r copies of 132 has an algebraic generating function (this was previously proved, constructively, by B… (More)

We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial.

In this survey paper, we explore the interactions between mathematics and engineering inspired by the challenge of transmitting data along powerlines. In particular, we focus on how combinatorial objects called permutation arrays offer a way of encoding data which allows the noise problems experienced in powerline communications (PLCs) to be overcome. The… (More)

Equidistant Frequency Permutation Arrays are combinatorial objects of interest in coding theory. A frequency permutation array is a type of constant composition code in which each symbol occurs the same number of times in each codeword. The problem is to find a set of codewords such that any pair of code-words are a given uniform Hamming distance apart. The… (More)

The existence of a primitive free (normal) cubic x 3 − ax 2 + cx − b over a finite field F with arbitrary specified values of a (= 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.