Combinatorial study and log-concavity of numerical sequences

Abstract

This thesis lies in the field of enumerative and algebraic combinatorics. It is devoted to the combinatorial and algebraic study of the sequences of Over-inversion numbers iB′ (n, k) and Mahonian numbers of type B iB(n, k), together with their q-analogue iBq (n, k), with a particular emphasis on their log-concavity and unimodality properties. We first introduce the concept of overlined permutations and define the associated Over-inversion numbers, for which we establish recurrence relations, fundamental identities, and a generating function. Several combinatorial interpretations are then provided in terms of lattice paths, overpartitions, and tilings, highlighting the deep combinatorial structure of these numbers. Next, we investigate Mahonian numbers of type B,providing recurrence relations, the Knuth–Netto formula, and generating functions for their subdiagonals on or below the main diagonal, as well as their combinatorial interpretations in terms of lattice paths, partitions, and tilings. These results are complemented by bijective interpretations involving lattice paths and signed permutations of the hyperoctahedral group Bn, showing how such permutations can be directly constructed from paths. A major contribution of this work is the introduction and analysis of a q-analogue of Mahonian numbers of type B, defined through a new inversion statistic on signed permutations. We explore its algebraic properties and provide combinatorial interpretations in terms of paths, partitions, and tilings. Finally, we establish combinatorial proofs of the log-concavity and unimodality of the Over-inversion numbers, and we prove that the q-analogue of Mahonian numbers of type B forms a strongly q-log-concave sequence of polynomials in k, which implies that the Mahonian numbers of type B form a log-concave sequence in k and therefore are unimodal.