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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2403.13686 |
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| _version_ | 1866913274888454144 |
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| author | Xu, Zijian |
| author_facet | Xu, Zijian |
| contents | A $k$-modal sequence is a sequence of real numbers that can be partitioned into $k+1$ (possibly empty) monotone sections such that adjacent sections have opposite monotonicities. For every positive integer $k$, we prove that any sequence of $n$ pairwise distinct real numbers contains a $k$-modal subsequence of length at least $\sqrt{(2k+1)(n-\frac14)} - \frac{k}{2}$, which is tight in a strong sense. This confirms an old conjecture of F.R.K.Chung (J.Combin.Theory Ser.A, 29(3):267-279, 1980). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $k$-modal subsequences Xu, Zijian Combinatorics 05D10 A $k$-modal sequence is a sequence of real numbers that can be partitioned into $k+1$ (possibly empty) monotone sections such that adjacent sections have opposite monotonicities. For every positive integer $k$, we prove that any sequence of $n$ pairwise distinct real numbers contains a $k$-modal subsequence of length at least $\sqrt{(2k+1)(n-\frac14)} - \frac{k}{2}$, which is tight in a strong sense. This confirms an old conjecture of F.R.K.Chung (J.Combin.Theory Ser.A, 29(3):267-279, 1980). |
| title | On $k$-modal subsequences |
| topic | Combinatorics 05D10 |
| url | https://arxiv.org/abs/2403.13686 |