Shape Avoiding Permutations

Permutations avoiding all patterns of a given shape (in the sense of Robinson-Schensted-Knuth) are considered. We show that the shapes of all such permutations are contained in a suitable thick hook, and deduce an exponential growth rate for their number.Comment: 16 pages; final form, to appear in J. Combin. Theory, Series

Young classes of permutations

We characterise those classes of permutations having the property that for every tableau shape either every permutation of that shape or no permutation of that shape belongs to the class. The characterisation is in terms of the dominance order for partitions (and their conjugates) and shows that for any such class there is a constant k such that no permutation in the class can contain both an increasing and a decreasing sequence of length k.Comment: 11 pages, this is the final version as accepted by the Australasian Journal of Combinatorics. Some more minor typos have been correcte

Grid classes and the Fibonacci dichotomy for restricted permutations

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

An Erd\H{o}s--Hajnal analogue for permutation classes

Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$

Pattern avoidance classes and subpermutations

Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers a structure theorem is given. The structure theorem shows that the class is almost closed under direct sums or has a rational generating function.Comment: 18 pages, 4 figures (all in-line