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

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 $\beta$.