We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

The problem of noiselessly encoding a message when prior statistics are known is considered. The close relationship of arithmetic and enumerative coding for this problem is shown by computing explicit arithmetic coding probabilities for various enumerative coding examples. This enables a comparison to be made of the coding efficiency of Markov models and enumerative codes as well as a new codin...

This paper presents a minimal enumerative approach to the problem of compiling typed unification grammars into CFG language models, a prototype implementation and results of experiments in which it was used to compile some non-trivial unification grammars. We argue that enumerative methods are considerably more useful than has been previously believed. Also, the simplicity of enumerative method...

The main objective of the thesis is to develop and evaluate a variety of computer-aided experimental methods that yield insight for discovering and proving theorems in combinatorics. We contribute to the methodology of the “creativity spiral” paradigm. We present nine studies, most of which rely on replacing equivalence of discrete structures by a finite group action. Throughout the thesis we m...

This thesis studies aspects of the set of acyclic orientations of a simple undirected graph. Acyclic orientations of a graph may be readily obtained from bijective labellings of its vertex-set with a totally ordered set, and they can be regarded as partially ordered sets. We will study this connection between acyclic orientations of a graph and the theory of linear extensions or topological sor...

The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying algebraically the structures thus obtained (changes of bases, generating sets, presentations, morphisms, representations), to collect combinatorial informatio...

Enumerative coding is an attractive algorithmic procedure for translating long source words into codewords and vice versa. The usage of long codewords makes it possible to approach a code rate which is as close as desired to Shannon’s noiseless capacity of the constrained channel. Enumerative encoding is prone to massive error propagation as a single bit error could ruin entire decoded words. T...

Hurwitz numbers count ramified covers of a Riemann surface with prescribed monodromy. As such, they are purely combinatorial objects. Tautological classes, on the other hand, are distinguished classes in the intersection ring of the moduli spaces of Riemann surfaces of a given genus, and are thus “geometric.” Localization computations in Gromov-Witten theory provide non-obvious relations betwee...