$p-$Ferrer diagram, $p-$linear ideals and arithmetical rank

Marcel Morales (1)

1 IF - Institut Fourier

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 \cite{m2}, which extends the notion of linearly joined ideals introduced and developped in the papers \cite{bm2}, \cite{bm4},\cite{eghp} and \cite{m1}. We can observe the connection between the results on this paper about the Poincaré series of a $p-$Ferrer diagram $\Phi $and the rook problem, which consist to put $k$ rooks in a non attacking position on the $p-$Ferrer diagram $\Phi $.

Mots clés: Ferrer-diagram; Poincaré series; Betti numbers; rook problem

Domaines: Algèbre commutative [math.AC]

Marcel Morales. $p-$Ferrer diagram, $p-$linear ideals and arithmetical rank. 2008. ⟨hal-00340278⟩