Sum expansion of the elementary symmetric polynomials

Question

Recently I stumbled upon the following equation:

$$e_k(V\cup W) = \sum_{i=0}^{|W|}e_{k-i}(V)e_i(W)$$

$V$ and $W$ are subsets of $\{x_0,x_1,...,x_n\}$, and $V\cap W = \ \varnothing$. Where both $e_k(V)$ and $e_{k-i}(W)$ are the elementary symmetric polynomials.

(Question) How would one prove this equation? My first attempt would be induction, but can it be done without induction?

Thanks in advance.

Answer 1

This follows since every $S \subseteq V \cup W$ can be decomposed uniquely into $S = S_1 \cup S_2$ where $S_1 \subseteq V$ and $S_2 \subseteq W$. Explicitly,

\begin{align*} e_k(V\cup W) &= \sum_{S \subseteq V \cup W, \, |S| = k} \prod_{x \in S} x\\ &= \sum_{S_1 \subseteq V, S_2 \subseteq W, |S_1 \cup S_2| = k} \prod_{x \in S_1} x\prod_{y \in S_2} y\\ &= \ldots\\ \end{align*}

(steps left you to fill.) Note that setting all $x_i = 1$ recovers Vandermonde's Identity.

Alternatively: Consider the product $$\prod_{x \in V \cup W} (1 + x) = \left( \prod_{x \in V}(1+x) \right) \left( \prod_{y \in W}(1+y) \right)$$

and consider the degree-$k$ homogeneous component of both sides.