ETD

• ab-index
• plethysm
• Poincarè extended ab-index

The starting point of this thesis is the recently introduced Poincaré-extended ab-index, which can be viewed as a common generalization of the classical ab-index and the Poincaré polynomial. One of the remarkable features of the extended ab-index is that, under suitable specializations, it recovers a variety of studied objects, as the flag Hilbert–Poincaré series associated with the Igusa zeta function of a hyperplane arrangement, as well as the Chow polynomial of a matroid. Stump showed that the extended ab-index provides a surprisingly elementary combinatorial framework in which one can recover the γ-positivity of the Chow polynomial of a matroid. In this way, the extended ab-index clarifies phenomena whose original proofs relied on sophisticated geometric methods. The purpose of this thesis is to study the Poincaré-extended ab-index from both combinatorial and algebraic perspectives: we present a combinatorial interpretation of its coefficients, proving that they are positive when the poset is R-labeled. We then discuss about a transformation which recovers the extended ab-index from the classic ab-index. Finally, we reinterpret this substitution in the framework of quasysimmetric functions with the help of plethysm.