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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.09240 |
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| _version_ | 1866908400666804224 |
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| author | Chan, Swee Hong Pak, Igor Panova, Greta |
| author_facet | Chan, Swee Hong Pak, Igor Panova, Greta |
| contents | We prove a weak version of the cross--product conjecture: ${F}(k+1,\ell) {F}(k,\ell+1) \geq (\frac12+\varepsilon) {F}(k,\ell) {F}(k+1,\ell+1)$, where ${F}(k,\ell)$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are $k$ and $\ell$ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the {generalized cross--product conjecture}. The proofs use geometric inequalities for mixed volumes and combinatorics of words. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09240 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the cross-product conjecture for the number of linear extensions Chan, Swee Hong Pak, Igor Panova, Greta Combinatorics Metric Geometry We prove a weak version of the cross--product conjecture: ${F}(k+1,\ell) {F}(k,\ell+1) \geq (\frac12+\varepsilon) {F}(k,\ell) {F}(k+1,\ell+1)$, where ${F}(k,\ell)$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are $k$ and $\ell$ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the {generalized cross--product conjecture}. The proofs use geometric inequalities for mixed volumes and combinatorics of words. |
| title | On the cross-product conjecture for the number of linear extensions |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2306.09240 |