The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns

We show the first known example for a pattern $q$ for which $\lim_{n\to \infty} \sqrt[n]{S_n(q)}$ is not an integer. We find the exact value of the limit and show that it is irrational. Then we generalize our results to an infinite sequence of patterns. Finally, we provide further generalizations that start explaining why certain patterns are easier to avoid than others. Finally, we show that if $q$ is a layered pattern of length $k$, then $L(q)\geq (k-1)^2$ holds.Comment: 10 pages, 1 figur

On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia

We construct a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley limit for the class of 4231-avoiding permutations is bounded below by 9.35. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Wilf-Stanley limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2.Comment: Submitted to Advances in Applied Mathematic

Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\ell$ is at most $4\ell^2$, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\ell$ is attained by a layered pattern then this implies an upper bound of $4\ell^2$ for the Stanley-Wilf limit of any pattern of length $\ell$. We also conjecture that, for any $k\ge 0$, the set of 1324-avoiding permutations with $k$ inversions contains at least as many permutations of length $n+1$ as those of length $n$. We show that if this is true then the Stanley-Wilf limit for 1324 is at most $e^{\pi\sqrt{2/3}} \simeq 13.001954$

Growth rates for subclasses of Av(321)

Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates

Asymptotic enumeration of permutations avoiding generalized patterns

Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns.Comment: 14 pages, 3 figures, to be published in Adv. in Appl. Mat