Projective dimension of powers of cover ideal of Ferrers graphs

Abstract

Let $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ be a partition with ${\lambda }_1=m$. Denote by $J_{\lambda }$ the cover ideal in the polynomial ring \( S = k[x_1, \ldots, x_n, y_1, \ldots, y_m] \) associated to the Ferrers graph corresponding to $\lambda$. Let $d(\lambda )$ denote the number of distinct parts of $\lambda$. We prove that \[ \operatorname{pd}(S/J_λ^t) = \min\{t,\; d(λ)\} + 1 \] for all $t\ge 1$.