The Poincaré-extended ab-index
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| author | Dorpalen-Barry, Galen Maglione, Joshua Stump, Christian |
| author_facet | Dorpalen-Barry, Galen Maglione, Joshua Stump, Christian |
| contents | Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended ab-index, which generalizes both the ab-index and the Poincaré polynomial. For posets admitting R-labelings, we give a combinatorial description of the coefficients of the extended ab-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 ab-index generalizing the cd-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-make a conjecture about Schur positivity. A proof of this conjecture now appears an appendix by Ricky Ini Liu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_05904 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Poincaré-extended ab-index Dorpalen-Barry, Galen Maglione, Joshua Stump, Christian Combinatorics 06A07, 05B35, 52C40 Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended ab-index, which generalizes both the ab-index and the Poincaré polynomial. For posets admitting R-labelings, we give a combinatorial description of the coefficients of the extended ab-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 ab-index generalizing the cd-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-make a conjecture about Schur positivity. A proof of this conjecture now appears an appendix by Ricky Ini Liu. |
| title | The Poincaré-extended ab-index |
| topic | Combinatorics 06A07, 05B35, 52C40 |
| url | https://arxiv.org/abs/2301.05904 |