Pattern Avoidance

A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of interest to this work. We first characterize the restricted growth functions of several avoidance classes based on partitions of size four, enumerate these avoidance classes, and consider the distribution of the Wachs and White statistics across these avoidance classes. We also investigate the equidistribution of statistics between avoidance classes based on multiple patterns

Avoidance of Partitions of a Three-element Set

Klazar defined and studied a notion of pattern avoidance for set partitions, which is an analogue of pattern avoidance for permutations. Sagan considered partitions which avoid a single partition of three elements. We enumerate partitions which avoid any family of partitions of a 3-element set as was done by Simion and Schmidt for permutations. We also consider even and odd set partitions. We provide enumerative results for set partitions restricted by generalized set partition patterns, which are an analogue of the generalized permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit of work done by Babson and Steingr{'{\i}}msson, we will show how these generalized partition patterns can be used to describe set partition statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied Mathematic

Mahonian STAT on words

In 2000, Babson and Steingr\'imsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In 2010, Burstein refined some of these results by giving a bijection between permutations with a fixed value for the major index and those with the same value for STAT, where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingr\'imsson's paper. Several other statistics are preserved as well by Burstein's bijection. At the Formal Power Series and Algebraic Combinatorics Conference (FPSAC) in 2010, Burstein asked whether his bijection has other interesting properties. In this paper, we not only show that Burstein's bijection preserves the Eulerian statistic ides, but also use this fact, along with the bijection itself, to prove Mahonity of the statistic STAT on words we introduce in this paper. The words statistic STAT introduced by us here addresses a natural question on existence of a Mahonian words analogue of STAT on permutations. While proving Mahonity of our STAT on words, we prove a more general joint equidistribution result involving two six-tuples of statistics on (dense) words, where Burstein's bijection plays an important role

Partout: A Distributed Engine for Efficient RDF Processing

The increasing interest in Semantic Web technologies has led not only to a rapid growth of semantic data on the Web but also to an increasing number of backend applications with already more than a trillion triples in some cases. Confronted with such huge amounts of data and the future growth, existing state-of-the-art systems for storing RDF and processing SPARQL queries are no longer sufficient. In this paper, we introduce Partout, a distributed engine for efficient RDF processing in a cluster of machines. We propose an effective approach for fragmenting RDF data sets based on a query log, allocating the fragments to nodes in a cluster, and finding the optimal configuration. Partout can efficiently handle updates and its query optimizer produces efficient query execution plans for ad-hoc SPARQL queries. Our experiments show the superiority of our approach to state-of-the-art approaches for partitioning and distributed SPARQL query processing