Let $n$ and $p$ be two positive integers. Consider the function $$\max_{n,p}:\{0,\dots,n\}^p\to\{0,\dots,n\}$$ that computes the maximum of a $p$-tuple of integers in the range $\{0,\dots,n\}$. Are there explicit expressions for symmetric polynomials $P_{n,p}\in\mathbb{Q}[\sigma_1,\dots,\sigma_p]$ such that $P_{n,p}(\sigma_1,\dots,\sigma_p)$ interpolates $\displaystyle\max_{n,p}$? Here the $\sigma_i$ are the elementary symmetric polynomials.

The case $p=2$ can be done by hand : $P_{n,2}(\sigma_1,\sigma_2)$ can be described by the formula $$\sum_{s=0}^{2n} \prod_{\substack{a=0\\a\neq s}} \frac{\sigma_1-a}{s-a} \cdot\left( \sum_{i=\max\{s-n,0\}}^{\lfloor s/2\rfloor} (s-i) \prod_{\substack{j=\max\{s-i,0\}\\j\neq i}}^{\lfloor s/2\rfloor} \frac{\sigma_2-j(s-j)}{i(s-i)-j(s-j)} \right)$$ Which you obtain by interpolating the maximum functions along the ``antidiagonals'' $x+y=s$, $0\leq x,y\leq n$.

As far as I can tell, the result from Interpolation for Symmetric Functions is inapplicable here.

I'm interested in this question to study networks. The special case where $n=p=2^N$ is of particular interest to me. Furthermore, I want to allow more invariant polynomials, specifically those that are invariant under a 2-Sylow group of $\mathfrak{S}_n$.