Combinatorial interpretations of elementary symmetric polynomials? I have some questions as to some good combinatorial interpretations for the sum of elementary symmetric polynomials. I know that for example, for n =2 we have that:
$e_0  = 1$
$e_1 = x_1+x_2$
$e_2 = x_1x_2$
And each of these can clearly been seen as the coefficient of $t^k$ in $(1+x_1t)(1+x_2t)$. Now, in general, what combinatorial interpreations are there for say:
$\sum_{i=0}^n e_i(x)$ for some $x = (x_1,...,x_n)$ ? 

Here are two specific interesting cases to go with Mariano Suárez-Alvarez's general explanation.  On $n$ variables, $e_k(1,1,\ldots,1) = \binom{n}{k}$ and $e_k(1,2,\ldots,n) = \left[ n+1 \atop n-k+1 \right]$, where the latter is a Stirling number of the first kind.  (See Comtet's Advanced Combinatorics, pp. 213-214.)  So summing the former over $k$ gives $2^n$, and summing the latter over $k$ gives $(n+1)!$.
