Let $$p\_k(x\_1,\ldots,x\_n)=x\_1^k+\cdots+x\_n^k$$ and let
$$e\_k(x\_1,\ldots,x\_n)=\sum\_{1\le i\_1<i\_2<\ldots<i\_k\le n}x\_{i\_1}\cdots x\_{i\_k} $$
be the $k$'th symmetric polynomial. It is well known that there exists a unique polynomial
$$ F\_k(y\_1,\ldots,y\_k)\in\mathbb Z[y\_1,\ldots,y\_k] $$
such that 
$$
p\_k(x\_1,\ldots,x\_n) = F\_k\bigl(e\_1(x\_1,\ldots,x\_n),\ldots,e\_k(x\_1,\ldots,x\_n)\bigr).$$
(See for example 
<https://en.wikipedia.org/w/index.php?title=Newton%27s_identities&oldid=733760875#Expressing_power_sums_in_terms_of_elementary_symmetric_polynomials>.) 


I have two questions about the polynomial $F\_k$.


1. Are there reasonable upper and lower bounds, in terms of $n$ and $k$, for the number of monomials that appear in $F\_k$?
2. Are there reasonable upper and lower bounds, in terms of $n$ and $k$, for the average absolute value of the coefficients of the monomials in $F\_k$?


And here a more specific question, which is where the above problems arose:


3. If we let $(\epsilon\_1,\ldots,\epsilon\_k)$ range over the $2^k$ elements of $\{-1,1\}^k$, find decent upper and lower bounds for the average value of $\bigl|F\_k(\epsilon\_1,\ldots,\epsilon\_k)\bigr|$ in terms of $n$ and $k$. In particular, is it true that if, say, $\frac14 n \le k \le \frac34 n$, then the average value of $\bigl|F\_k(\epsilon\_1,\ldots,\epsilon\_k)\bigr|$ grows exponentially in $n$?



Coefficients of polynomials appearing in Newton identities relating symmetric polynomials to power sums
OEIS A00041 gives the number of monomials. The absolute values of the coefficients of the polynomials sum to $2^k-1$. See A263916 for more notes and references on these Faber polynomials.


