A note on the Björner--Kalai theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908692205535232 |
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| author | Zhan, Xiongfeng Huang, Xueyi |
| author_facet | Zhan, Xiongfeng Huang, Xueyi |
| contents | In 1988, Björner and Kalai used combinatorial shadow functions to characterize the maximal Betti sequence for a given $f$-vector and the minimal $f$-vector for a given Betti sequence. Their description of the maximal Betti sequence was expressed through a set of inequalities. In this paper, we introduce an error function $δ_k$ associated with the combinatorial shadow functions and use it to sharpen these inequalities into exact equalities. As a corollary, we obtain an equivalent form of Björner and Kalai's characterization of all possible pairs $(f,β)$ that can occur as the $f$-vector and Betti sequence of a simplicial complex. Moreover, combining our results with a previous result of Björner in 2011, we derive a new number-theoretic inequality concerning the count of odd square-free integers with a specified number of prime factors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05943 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the Björner--Kalai theorem Zhan, Xiongfeng Huang, Xueyi Combinatorics 05E45 In 1988, Björner and Kalai used combinatorial shadow functions to characterize the maximal Betti sequence for a given $f$-vector and the minimal $f$-vector for a given Betti sequence. Their description of the maximal Betti sequence was expressed through a set of inequalities. In this paper, we introduce an error function $δ_k$ associated with the combinatorial shadow functions and use it to sharpen these inequalities into exact equalities. As a corollary, we obtain an equivalent form of Björner and Kalai's characterization of all possible pairs $(f,β)$ that can occur as the $f$-vector and Betti sequence of a simplicial complex. Moreover, combining our results with a previous result of Björner in 2011, we derive a new number-theoretic inequality concerning the count of odd square-free integers with a specified number of prime factors. |
| title | A note on the Björner--Kalai theorem |
| topic | Combinatorics 05E45 |
| url | https://arxiv.org/abs/2504.05943 |