A class of square-free monomial ideals associated to two integer sequences

Given two finite sequences of positive integers $\alpha$ and $\beta$, we associate a square free monomial ideal $I_{\alpha ,\beta }$ in a ring of polynomials $S$, and we recursively compute the algebraic invariants of $S\mathrm{/}{I}_{\alpha ,\beta }$. Also, we give precise formulas in special cases.