Exercise 8.8 in Monomial Ideals by Herzog and Hibi:
Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators $f_{1},...,f_{m}$ with deg$(f_{1})\leq...$ deg$(f_{m})$. Prove that
$\beta_{i,i+j}(I)=\sum\limits_{k=1,deg(f_{k})=j}^{m}$ $\left( \begin{array}{c} r_{k}\\ i \end{array} \right)$
where $r_{k}$ is the cardinality of the minimal system of linear forms which generate $(f_{1},...,f_{k-1}):f_{k}$.
The case when deg$f_{1}=$deg$f_{2}=...=$deg$f_{m}$ may be proved easily because in this case $I$ has linear resolution.
When degrees are not equal, how to prove? We know from Theorem 8.2.15 in this book that $I$ is componentwise linear.
