Several New Generalizations of LYM Inequality
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914394391183360 |
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| author | Huang, Zihao Liang, Weikang Ma, Yujiao Wang, Suijie |
| author_facet | Huang, Zihao Liang, Weikang Ma, Yujiao Wang, Suijie |
| contents | The LYM inequality is a fundamental result concerning the sizes of subsets in a Sperner family. Subsequent studies on the LYM inequality have been generalized to families of $r$-decompositions, where all components are required to avoid chains of the same length. In this paper, we relax this constraint by allowing components of a family of $r$-decompositions to avoid chains of distinct lengths, and derive generalized LYM inequalities across all the relevant settings, including set-theoretic, $q$-analog, continuous analog, and arithmetic analog frameworks. Notably, the bound in our LYM inequalities does not depend on the maximal length of all forbidden chains. Moreover, we extend our approach beyond $r$-decompositions to $r$-multichains, and establish analogous LYM inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21024 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Several New Generalizations of LYM Inequality Huang, Zihao Liang, Weikang Ma, Yujiao Wang, Suijie Combinatorics The LYM inequality is a fundamental result concerning the sizes of subsets in a Sperner family. Subsequent studies on the LYM inequality have been generalized to families of $r$-decompositions, where all components are required to avoid chains of the same length. In this paper, we relax this constraint by allowing components of a family of $r$-decompositions to avoid chains of distinct lengths, and derive generalized LYM inequalities across all the relevant settings, including set-theoretic, $q$-analog, continuous analog, and arithmetic analog frameworks. Notably, the bound in our LYM inequalities does not depend on the maximal length of all forbidden chains. Moreover, we extend our approach beyond $r$-decompositions to $r$-multichains, and establish analogous LYM inequalities. |
| title | Several New Generalizations of LYM Inequality |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.21024 |