2024/12/20 by Galen Dorpalen‐Barry, Joshua Maglione, Christian Stump · 1 voice
Mathematics · #Advanced Combinatorial Mathematics #Mathematics and Applications #Functional Equations Stability Results
paper · doi:10.1112/jlms.70054
openalex publication_date 2024/12/20 · openalex created_date 2024/12/21 · openalex updated_date 2026/05/21
Abstract Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré‐extended ‐index , which generalizes both the ‐index and the Poincaré polynomial. For posets admitting ‐labelings, we give a combinatorial description of the coefficients of the extended ‐index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and Kühne. We also define the pullback ‐index, generalizing the ‐index of face posets for oriented matroids. Our results recover, generalize, and unify results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and — in the special case of symmetric functions — pose a conjecture about Schur positivity. This conjecture was strengthened and proved by Ricky Liu, and the proof appears as an appendix.