Number of $k$-normal elements over a finite field

An element $\alpha \in \mathbb{F}_{q^n}$ is a normal element over $\mathbb{F}_q$ if the conjugates $\alpha^{q^i}$, $0 \leq i \leq n-1$, are linearly independent over $\mathbb{F}_q$. Hence a normal basis for $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ is of the form $\{\alpha,\alpha^q, \ldots, \alpha^{q^{n-1}}\}$, where $\alpha \in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$. In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of k-normal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of $k$-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).