Some remarks on the Betti numbers of squarefree ideals

Let K be a field, $\Delta$ a simplicial complex on the vertex set [n] = {1, ..., n}and $I_\Delta$ the Stanley-Reisner ideal of $\Delta$ over K. If $\Delta$ is a lexsegment simplicial complex, that is $I_\Delta$ is a lexsegment ideal of the polynomial ring $K[X_1, ...,X_n]$, we state new formulas for the minimal number of generators of $I_\Delta$ and for the Betti numbers of $I_\Delta$.