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arXiv · 2204.03033

Repeatable patterns and the maximum multiplicity of a generator in a reduced word

Christian Gaetz·Yibo Gao·Pakawut Jiradilok·Gleb Nenashev·Alexander Postnikov
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Abstract

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.

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Christian Gaetz, Yibo Gao, Pakawut Jiradilok, Gleb Nenashev, Alexander Postnikov. 2022-04-06. Repeatable patterns and the maximum multiplicity of a generator in a reduced word. https://doi.org/10.5070/c64264252

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