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

TL;DR

This paper introduces p-Ferrer diagrams and associated ideals, analyzing their algebraic properties such as regularity and Betti numbers, and explores connections to combinatorial rook placements.

Contribution

It defines p-Ferrer diagrams and ideals, proves their Castelnuovo-Mumford regularity is p+1, and studies their resolutions and Betti numbers, extending the concept of linearly joined ideals.

Findings

p-Ferrer ideals have Castelnuovo-Mumford regularity p+1

Betti numbers and minimal resolutions of p-Ferrer ideals are characterized

Connection established between Poincaré series of p-Ferrer diagrams and rook placement problem

Abstract

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 Poincar\'e 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 $.