For the elementary symmetric function $$e_k(\mathbf{x}) = \sum_{\substack{\mathbf{y} : y_j \in {0,1},\\\sum_j y_j = k}} \prod_{j=1}^n x_j^{y_j}$$
for a fixed $\mathbf{x}$ where all elements are positive if I plot the $e_k$ for $k=1,\ldots,n$ where $n$ is length of $\mathbf{x}$ I see these beautiful unimodal log-concave shaped curves:
I'm wondering if there is any work related to approximating this curve or perhaps calculating moments