Average-Rare Order Ideals in Functional Preorders
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911284542308352 |
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| author | Hachimori, Masahiro Kashiwabara, Kenji |
| author_facet | Hachimori, Masahiro Kashiwabara, Kenji |
| contents | We prove that for the preorder induced by a function f: V -> V, the family of all order ideals is average-rare, that is, its normalized degree sum (nds) is nonpositive. As a base case in our reduction, we establish the same result for functional partial orders (or rooted forests). We also propose a conjecture related to Frankl's Conjecture. All proofs have been formally verified in the proof assistant Lean 4. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Average-Rare Order Ideals in Functional Preorders Hachimori, Masahiro Kashiwabara, Kenji Combinatorics 05D05 We prove that for the preorder induced by a function f: V -> V, the family of all order ideals is average-rare, that is, its normalized degree sum (nds) is nonpositive. As a base case in our reduction, we establish the same result for functional partial orders (or rooted forests). We also propose a conjecture related to Frankl's Conjecture. All proofs have been formally verified in the proof assistant Lean 4. |
| title | Average-Rare Order Ideals in Functional Preorders |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2511.19833 |