Computer-assisted Studies in Algebraic Combinatorics

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 make much use of computer algebra systems. In particular, the first core chapter is completely devoted to an improvement of an algorithm that frequently occurs in proving combinatorial identities, namely Gosper’s algorithm. The enumerative part of the thesis is centered around the concept of quasi-polynomials. We show that many interesting combinatorial quantities, typically depending on two parameters, possess a quasipolynomial closed form if one of the parameters becomes fixed. Then we derive an algorithm for computing the values of one special kind of quasi-polynomials, namely the number of restricted partitions. In the constructive part of the thesis we subsequently focus our attention on different kinds of discrete structures. We start with two chapters on graphs, disproving a graph-theoretical conjecture in the first one and extending the classification theory of chordal rings in the second one. Then we switch to necklaces and bracelets. We develop an algorithm that generates bracelets and then we improve known bounds on the relation between local and global bead proportionalities in bracelets. The rest of the constructive part is devoted to finite geometries and linear codes. We build catalogs of two kinds of configurations in projective planes over finite fields, namely the semiovals and the arcs. We develop general constructions of semiovals in Desarguesian planes of arbitrary odd orders. At the end we classify certain optimal ternary linear codes. The majority of our studies improves or extends results of other authors. This was achieved by thoroughly planned interlacing of human thinking steps and machine computing. Hence, our main conclusion is that this strategy inevitably contributes to the research in combinatorics.