Suppose that $\alpha_1, \cdots, \alpha_n$ be $n$ roots of the polynomial equation $p(x)=0$ of degree $n$. I was studying on symmetric polynomial and have come accross of several problems on like $\sum \alpha_i^2, \sum \alpha_i^3\alpha_j^2$ etc in terms of the coefficients of $p(x)$. Then the following question came to mind.
Suppose we consider $\sum \alpha_1^{a_1}\alpha_2^{a_2}\cdots \alpha_r^{a_r}\alpha_{r+1}\alpha_{r+2}\cdots \alpha_s$ where $a_1, \cdots, a_r>1$ and $s\leq n$. In other words, we are considering the sum of all the terms of the above form where the first $r$ $\alpha$s are of higher power and the rest are of unit power.
My question is: how to find out how many terms will be there in this sum? Is it possible to get the answer in closed form?
Thanks in advance
The sum you are considering is called the monomial symmetric function $m(a_1,a_2,\ldots,a_r,1,\ldots,1)$. One term is determined by associating a specific exponent (possibly $0$) to each root $\alpha_i$, in such a way that each exponent is used exactly the right number of times. In other words you can consider the available exponents (again including occurrences of $0$) as $n$ letters, to be used in a word formed by rearranging them (as in Scrabble), called the multiset permutations of the multiset of letters.
The number of (distinct) multiset permutations of a given finite multiset depends only on the multiplicities $m_1,\ldots,m_k$ of the various letters that occur (not on what those letters are). It is a multinomial coefficient $$ \binom n{m_1,\ldots,m_k} \quad\text{where $n=m_1+\cdots+m_k$.} $$