For a subset $I$ of $[n]$, a hyperplane $H_I \subset \mathbb{R}^n$ is defined by $$\sum_{i \in I} x_i= \sum_{j \not\in I} x_j.$$ Have you seen the following hyperplane arrangements? Is there anything interesting?

• $n=2k$, $\mathcal{A}=\{H_I \mid |I|=k\}$

• $n \ge 2k > 0$, $\mathcal{A}=\{H_I \mid k \le |I| \le n-k\}$