Several combinatorial inequalities related to squarefree monomial ideals

Let $K$ be a field and $S=K[x_1,\dots ,x_n]$, 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 $0\subset I⊊J\subset S$, we prove several combinatorial inequalities which involve the coefficients of the polynomial $f(t)=(1+t+\cdots +t^{m-1}{)}^n$.