4 research outputs found
Permutation Patterns, Reduced Decompositions with Few Repetitions and the Bruhat Order
This thesis is concerned with problems involving permutations. The main focus is on connections between permutation patterns and reduced decompositions with few repetitions. Connections between permutation patterns and reduced decompositions were first studied various mathematicians including Stanley, Billey and Tenner. In particular, they studied pattern avoidance conditions on reduced decompositions with no repeated elements. This thesis classifies the pattern avoidance and containment conditions on reduced decompositions with one and two elements repeated. This classification is then used to obtain new enumeration results for pattern classes related to the reduced decompositions and introduces the technique of counting pattern classes via reduced decompositions. In particular, counts on pattern classes involving 1 or 2 copies of the patterns 321 and 3412 are obtained. Pattern conditions are then used to classify and enumerate downsets in the Bruhat order for the symmetric group and the rook monoid which is a generalization of the symmetric group. Finally, motivated by coding theory, the concepts of displacement, additive stretch and multiplicative stretch of permutations are introduced. These concepts are then analyzed with respect to maximality and distribution as a new prospect for improving interleaver design
Permutations, moments, measures
We present a continued fraction with 13 permutation statistics, several of them new, connecting a great number of combinatorial structures to a wide variety of moment sequences and their measures from classical and noncommutative probability. The Hankel determinants of these moment sequences are a product of (p,q)-factorials, unifying several instances from the literature. The corresponding measures capture as special cases several classical laws, such as the Gaussian, Poisson, and exponential, along with further specializations of the orthogonalizing measures in the q-Askey scheme and several known noncommutative central limits. Statistics in our continued fraction generalize naturally to signed and colored permutations, and to the k-arrangements introduced here, permutations with k-colored fixed points
Combinatorics of the Permutahedra, Associahedra, and Friends
I present an overview of the research I have conducted for the past ten years
in algebraic, bijective, enumerative, and geometric combinatorics. The two main
objects I have studied are the permutahedron and the associahedron as well as
the two partial orders they are related to: the weak order on permutations and
the Tamari lattice. This document contains a general introduction (Chapters 1
and 2) on those objects which requires very little previous knowledge and
should be accessible to non-specialist such as master students. Chapters 3 to 8
present the research I have conducted and its general context. You will find:
* a presentation of the current knowledge on Tamari interval and a precise
description of the family of Tamari interval-posets which I have introduced
along with the rise-contact involution to prove the symmetry of the rises and
the contacts in Tamari intervals;
* my most recent results concerning q, t-enumeration of Catalan objects and
Tamari intervals in relation with triangular partitions;
* the descriptions of the integer poset lattice and integer poset Hopf
algebra and their relations to well known structures in algebraic
combinatorics;
* the construction of the permutree lattice, the permutree Hopf algebra and
permutreehedron;
* the construction of the s-weak order and s-permutahedron along with the
s-Tamari lattice and s-associahedron.
Chapter 9 is dedicated to the experimental method in combinatorics research
especially related to the SageMath software. Chapter 10 describes the outreach
efforts I have participated in and some of my approach towards mathematical
knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche