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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2022
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| Online-Zugang: | https://arxiv.org/abs/2204.03033 |
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| _version_ | 1866912055909416960 |
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| author | Gaetz, Christian Gao, Yibo Jiradilok, Pakawut Nenashev, Gleb Postnikov, Alexander |
| author_facet | Gaetz, Christian Gao, Yibo Jiradilok, Pakawut Nenashev, Gleb Postnikov, Alexander |
| contents | We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation $w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous work on sorting networks and on the "$k$-set" problem. After reinterpreting the problem in terms of monotone weakly separated paths, we show that, for fixed $k$ and sufficiently large $n$, the optimal density is realized by paths which are periodic in a precise sense, so that \[ \mathcal{M}(k,n)=c_k n + p_k(n) \] for a periodic function $p_k$ and constant $c_k$. In fact we show that $c_k$ is always rational, and compute several bounds and exact values for this quantity with "repeatable patterns", which we introduce. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_03033 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Repeatable patterns and the maximum multiplicity of a generator in a reduced word Gaetz, Christian Gao, Yibo Jiradilok, Pakawut Nenashev, Gleb Postnikov, Alexander Combinatorics We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation $w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous work on sorting networks and on the "$k$-set" problem. After reinterpreting the problem in terms of monotone weakly separated paths, we show that, for fixed $k$ and sufficiently large $n$, the optimal density is realized by paths which are periodic in a precise sense, so that \[ \mathcal{M}(k,n)=c_k n + p_k(n) \] for a periodic function $p_k$ and constant $c_k$. In fact we show that $c_k$ is always rational, and compute several bounds and exact values for this quantity with "repeatable patterns", which we introduce. |
| title | Repeatable patterns and the maximum multiplicity of a generator in a reduced word |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2204.03033 |