Permutations Avoiding Arithmetic Patterns

Permutations Avoiding Arithmetic Patterns

A permutation π of an abelian group G (that is, a bijection from G to itself) will be said to avoid arithmetic progressions if there does not exist any triple (a, b, c) of elements of G, not all equal, such that c − b = b − a and π(c) − π(b) = π(b) − π(a). The basic question is, which abelian groups possess such a permutation? This and problems of a similar nature will be considered. 1 Notation...

Permutations Avoiding Arithmetic Progressions

On permutations avoiding arithmetic progressions

Let S be a subset of the positive integers, and let σ be a permutation of S. We say that σ is a k-avoiding permutation of S if σ does not contain any k-term AP as a subsequence. Similarly, the set S is said to be k-avoidable if there exists a k-avoiding permutation of S.

Symmetric Permutations Avoiding Two Patterns ∗

Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D4, the symmetry group of a square. We examine pattern-avoiding permutations with 180◦ rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our resu...

Avoiding patterns in irreducible permutations

We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index i such that σ(i + 1) − σ(i) = 1. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary nu...