  Let $K$ be a field and $S=K[x_1,\ldots,x_n]$, the ring of polynomials in $n$
variables, over $K$. Using the fact that the Hilbert depth is an upper bound
for the Stanley depth of a quotient of squarefree monomial ideals $0\subset
I\subsetneq J\subset S$, we prove several combinatorial inequalities which
involve the coefficients of the polynomial $f(t)=(1+t+\cdots+t^{m-1})^n$.
