I want to prove the following is an integer
$\frac{m(r_1 + r_2 + \cdots + r_m - 1)!}{r_1! r_2! \cdots r_m!}$
where $r_i$ are non-negative integers and they satisfy $r_1 + 2r_2 + \cdots +mr_m = m$.
Actually, those are the coefficients of the relation between the power sums and the elementary symmetric polynomials in the Newton-Girard identity.
How can I see the above combinatoric quantity is an integer?
$$\frac{m(r_1+r_2+\cdots+r_m-1)!}{r_1!r_2!\cdots r_m!}$$ $$=\sum_{i=1}^m\frac{i(r_1+r_2+\cdots+r_m-1)!}{r_1!r_2!\cdots(r_i-1)!\cdots r_m!}.$$ Each term of the above sum is an integer, so is the sum.