Schubert Numbers

This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders

Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints

In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from a recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polynomials, we describe a bijection between the class of k-parallelogram polyominoes and the class of planted planar trees having height less than k+3. In the second part of the thesis we examine the notion of pattern avoidance, which has been extensively studied for permutations. We introduce the concept of pattern avoidance in the context of matrices, more precisely permutation matrices and polyomino matrices. We present definitions analogous to those given for permutations and in particular we define polyomino classes, i.e. sets downward closed with respect to the containment relation. So, the study of the old and new properties of the redefined sets of objects has not only become interesting, but it has also suggested the study of the associated poset. In both approaches our results can be used to treat open problems related to polyominoes as well as other combinatorial objects.Comment: PhD thesi

Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot-Jacobi determinant identity with the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.Comment: 62 pages; v3 slightly updated relative to published versio

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure

Combinatorics, Probability and Computing

One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs