Given a finite poset $\mathbb{P}$ define $\varphi(\mathbb{P})$ to be the maximum value the Möbius function takes on an interval of $\mathbb{P}$. Given a natural number $n$ define $\varphi(n)$ to be the maximum value $\varphi(\mathbb{P})$ takes over all posets $\mathbb{P}$ of cardinality $n$.
I must show that $\varphi(n)>n^2$ and $\varphi(n) < n^n$ for sufficiently large $n$.
By Möbius function I mean the element of the incidence algebra given by $$\mu(x,y)=-\sum_{x\leq z < y} \mu(x,z)$$ which is the inverse of the $\zeta$ function.
I've been trying to construct examples of posets $\mathbb{P}$ such that I can find an interval where $\mu$ is bigger than $|\mathbb{P}|^2$ but have had no luck, the best I've done is construct a poset such that $\mu(\min,\max)$ is $|\mathbb{P}|-1$ ($1$ minimum $1$ maximum and $|\mathbb{P}|-2$ intermediate pairwise incompatible points). On the other hand I have no idea how to tackle $\varphi(n) < n^n$.
I appreciate any kind of help. Thank you for reading.