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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.17683 |
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| _version_ | 1866908722914131968 |
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| author | Kanungo, Shihan |
| author_facet | Kanungo, Shihan |
| contents | In this paper, we study the saturation function $\mathrm{Sat}(n,u)$ for sequences. Saturation for sequences was introduced by Anand, Geneson, Kaustav, and Tsai (2021), who proved that $\mathrm{Sat}(n,u)=O(n)$ for two-letter sequences $u$ and conjectured that this bound holds for all sequences. We present an algorithm that constructs a $u$-saturated sequence on $n$ letters and apply it to show $\mathrm{Sat}(n,u)=O(n)$ for several families of sequences $u$, including all repetitions of the form $abcabc\dots$. We further establish $\mathrm{Sat}(n,u)=O(n)$ for a broad class of sequences of the form $aa\dots bb$. In addition, we prove that for most sequences $u$, there exists an infinite $u$-saturated sequence. For three-letter sequences of the form $abc\dots xyz$, where $a,b,c$ are distinct and $xyz$ is a permutation of $abc$, we show -- under certain structural assumptions on $u$ -- that $\mathrm{Sat}(n,u)=O(n)$. Finally, we describe a linear program that computes the exact value of $\mathrm{Sat}(n,u)$ for arbitrary $n$ and $u$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper Bounds for Sequence Saturation Kanungo, Shihan Combinatorics 05D99 In this paper, we study the saturation function $\mathrm{Sat}(n,u)$ for sequences. Saturation for sequences was introduced by Anand, Geneson, Kaustav, and Tsai (2021), who proved that $\mathrm{Sat}(n,u)=O(n)$ for two-letter sequences $u$ and conjectured that this bound holds for all sequences. We present an algorithm that constructs a $u$-saturated sequence on $n$ letters and apply it to show $\mathrm{Sat}(n,u)=O(n)$ for several families of sequences $u$, including all repetitions of the form $abcabc\dots$. We further establish $\mathrm{Sat}(n,u)=O(n)$ for a broad class of sequences of the form $aa\dots bb$. In addition, we prove that for most sequences $u$, there exists an infinite $u$-saturated sequence. For three-letter sequences of the form $abc\dots xyz$, where $a,b,c$ are distinct and $xyz$ is a permutation of $abc$, we show -- under certain structural assumptions on $u$ -- that $\mathrm{Sat}(n,u)=O(n)$. Finally, we describe a linear program that computes the exact value of $\mathrm{Sat}(n,u)$ for arbitrary $n$ and $u$. |
| title | Upper Bounds for Sequence Saturation |
| topic | Combinatorics 05D99 |
| url | https://arxiv.org/abs/2512.17683 |