On the growth of the M\"obius function of permutations

TL;DR

This paper investigates the growth of the Möbius function in the permutation containment poset, presenting a sequence of permutations with polynomially growing Möbius values, thus improving previous bounds.

Contribution

The authors construct permutations with Möbius function values growing as a degree 7 polynomial, establishing the fastest known growth rate in relation to permutation size.

Findings

Constructed permutations with polynomial Möbius function growth

Established the fastest known growth rate of |μ(1,π)|

Provided a formula relating Möbius function to permutation embeddings

Abstract

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