P−ferrer Diagram, P−linear Ideals and Arithmetical Rank

In this paper we introduce p−Ferrer diagram, note that 1− Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any p−Ferrer diagram we associate a p−Ferrer ideal. We prove that p−Ferrer ideal have Castelnuovo mumford regularity p + 1. We also study Betti numbers , minimal resolutions of p−Ferrer ideals. Every p−Ferrer ideal is p−joined ideals in a sense defined in a fortcoming paper [M], which extends the notion of linearly joined ideals introduced and developped in the papers [BM2], [BM4],[EGHP] and [M]. We can observe the connection between the results on this paper about the Poincaré series of a p−Ferrer diagram Φand the rook problem, which consist to put k rooks in a non attacking position on the p−Ferrer diagram Φ.