form, stability and behavior of the solutions of a variety of systems of differences equations

Abstract

This thesis belongs to the field of combinatorics, or to be more accurate the enumerative combinatorics and bijective combinatorics. It addresses the study of log6concavity and log-convexity of certain sequences and sequences of polynomials by a combinatorial approach in which we use the combinatorial interpretation of each one. We define first an overpartition analogues of bisnomial coefficients, and we interpret them by overpartitions and by generalized Delannoy paths. Through these interpretations we prove the property of strong log-concavity. Next, we give a q-analogue of the number of permutations with a fixed number of inversions known as "Mahonian numbers". We give the link with partitions and paths, and hence prove the log-concavity. Finally, we study the two-Catalan triangle and in particular the two-Catalan numbers. We establish the combinatorial interpretation by a subset of the set of vertically constrained Motzkin-like paths, thereafter we prove the log-convexity of two-Catalan numbers and the log-concavity of rows of two-Catalan triangle.