568 research outputs found
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
A Formal Definition for Configuration
There exists a wide set of techniques to perform keyword-based search over
relational databases but all of them match the keywords in the users' queries
to elements of the databases to be queried as first step. The matching process
is a time-consuming and complex task. So, improving the performance of this
task is a key issue to improve the keyword based search on relational data
sources.In this work, we show how to model the matching process on
keyword-based search on relational databases by means of the symmetric group.
Besides, how this approach reduces the search space is explained in detail
Combinatorial Variations on Cantor's Diagonal
We discuss counting problems linked to finite versions of Cantor's diagonal
of infinite tableaux. We extend previous results of [2] by refining an
equivalence relation that reduces significantly the exhaustive generation. New
enumerative results follow and allow to look at the sub-class of the so- called
bi-Cantorian tableaux. We conclude with a correspondence between Cantorian-type
tableaux and coloring of hypergraphs having a square number of vertices
Shifted symmetric functions and multirectangular coordinates of Young diagrams
In this paper, we study shifted Schur functions , as well as a
new family of shifted symmetric functions linked to Kostka
numbers. We prove that both are polynomials in multi-rectangular coordinates,
with nonnegative coefficients when written in terms of falling factorials. We
then propose a conjectural generalization to the Jack setting. This conjecture
is a lifting of Knop and Sahi's positivity result for usual Jack polynomials
and resembles recent conjectures of Lassalle. We prove our conjecture for
one-part partitions.Comment: 2nd version: minor modifications after referee comment
Asymptotic behavior of some statistics in Ewens random permutations
The purpose of this article is to present a general method to find limiting
laws for some renormalized statistics on random permutations. The model
considered here is Ewens sampling model, which generalizes uniform random
permutations. We describe the asymptotic behavior of a large family of
statistics, including the number of occurrences of any given dashed pattern.
Our approach is based on the method of moments and relies on the following
intuition: two events involving the images of different integers are almost
independent.Comment: 32 pages: final version for EJP, produced by the author. An extended
abstract of 12 pages, published in the proceedings of AofA 2012, is also
available as version
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