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

A code for square permutations and convex permutominoes

In this article we consider square permutations, a natural subclass of permutations defined in terms of geometric conditions, that can also be described in terms of pattern avoiding permutations, and convex permutoninoes, a related subclass of polyominoes. While these two classes of objects arised independently in various contexts, they play a natural role in the description of certain random horizontally and vertically convex grid configurations. We propose a common approach to the enumeration of these two classes of objets that allows us to explain the known common form of their generating functions, and to derive new refined formulas and linear time random generation algorithms for these objects and the associated grid configurations.Comment: 18 pages, 10 figures. Revision according to referees' remark

Growth rates of permutation classes: categorization up to the uncountability threshold

In the antecedent paper to this it was established that there is an algebraic number $\xi\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $\xi$, there are only countably many less than $\xi$. Here we provide a complete characterization of the growth rates less than $\xi$. In particular, this classification establishes that $\xi$ is the least accumulation point from above of growth rates and that all growth rates less than or equal to $\xi$ are achieved by finitely based classes. A significant part of this classification is achieved via a reconstruction result for sum indecomposable permutations. We conclude by refuting a suggestion of Klazar, showing that $\xi$ is an accumulation point from above of growth rates of finitely based permutation classes.Comment: To appear in Israel J. Mat

goSLP: Globally Optimized Superword Level Parallelism Framework

Modern microprocessors are equipped with single instruction multiple data (SIMD) or vector instruction sets which allow compilers to exploit superword level parallelism (SLP), a type of fine-grained parallelism. Current SLP auto-vectorization techniques use heuristics to discover vectorization opportunities in high-level language code. These heuristics are fragile, local and typically only present one vectorization strategy that is either accepted or rejected by a cost model. We present goSLP, a novel SLP auto-vectorization framework which solves the statement packing problem in a pairwise optimal manner. Using an integer linear programming (ILP) solver, goSLP searches the entire space of statement packing opportunities for a whole function at a time, while limiting total compilation time to a few minutes. Furthermore, goSLP optimally solves the vector permutation selection problem using dynamic programming. We implemented goSLP in the LLVM compiler infrastructure, achieving a geometric mean speedup of 7.58% on SPEC2017fp, 2.42% on SPEC2006fp and 4.07% on NAS benchmarks compared to LLVM's existing SLP auto-vectorizer.Comment: Published at OOPSLA 201