Intersecting Families of Permutations

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.Comment: 'Erratum' section added. Yuval Filmus has recently pointed out that the 'Generalised Birkhoff theorem', Theorem 29, is false for k > 1, and so is Theorem 27 for k > 1. An alternative proof of the equality part of the Deza-Frankl conjecture is referenced, bypassing the need for Theorems 27 and 2

Representations From Group Actions On Words And Matrices

We provide a combinatorial interpretation of the frequency of any irreducible representation of Sn in representations of Sn arising from group actions on words. Recognizing that representations arising from group actions naturally split across orbits yields combinatorial interpretations of the irreducible decompositions of representations from similar group actions. The generalization from group actions on words to group actions on matrices gives rise to representations that prove to be much less transparent. We share the progress made thus far on the open problem of determining the irreducible decomposition of certain representations of Sm × Sn arising from group actions on matrices

Isomerism as Manifestation of Intrinsic Symmetry of Molecules: Lunn–Senior’s Theory

This article presents the principal results of the doctoral thesis “Isomerism as internal symmetry of molecules” by Valentin Vankov Iliev (Institute of Mathematics and Informatics), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 15 December, 2008.This paper is an extended review of our doctoral thesis “Isomerism as Intrinsic Symmetry of Molecules” in which we present, continue, generalize, and trace out Lunn–Senior’s theory of isomerism in organic chemistry

Symmetry reduction in convex optimization with applications in combinatorics

This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics

Of matroid polytopes, chow rings and character polynomials

Matroids are combinatorial structures that capture various notions of independence. Recently there has been great interest in studying various matroid invariants. In this thesis, we study two such invariants: Volume of matroid base polytopes and the Tutte polynomial. We gave an approach to computing volume of matroid base polytopes using cyclic flats and apply it to the case of sparse paving matroids. For the Tutte polynomial, we recover (some of) its coefficients as degrees of certain forms in the Chow ring of underlying matroid. Lastly, we study the stability of characters of the symmetric group via character polynomials. We show a combinatorial identity in the ring of class functions that implies stability results for certain class of Kronecker coefficients

ANALYTIC AND TOPOLOGICAL COMBINATORICS OF PARTITION POSETS AND PERMUTATIONS

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern, we show how to find the equation determining the spectrum. We give two length 4 applications, and a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible