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arXiv · 1809.05774

On the growth of the Möbius function of permutations

Vít Jelínek·Ida Kantor·Jan Kynčl·Martin Tancer
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Abstract

We study the values of the Möbius function $μ$ of intervals in the containment poset of permutations. We construct a sequence of permutations $π_n$ of size $2n-2$ for which $μ(1,π_n)$ is given by a polynomial in $n$ of degree 7. This construction provides the fastest known growth of $|μ(1,π)|$ in terms of $|π|$, improving a previous quadratic bound by Smith. Our approach is based on a formula expressing the Möbius function of an arbitrary permutation interval $[α,β]$ in terms of the number of embeddings of the elements of the interval into $β$.

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Vít Jelínek, Ida Kantor, Jan Kynčl, Martin Tancer. 2019-08-06. On the growth of the Möbius function of permutations. https://doi.org/10.1016/j.jcta.2019.105121

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