On the $(n,d{)}^{th}$ $f$-Ideals

Abstract

A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both ${\delta }_{\mathcal{F}}(I)$ and ${\delta }_{\mathcal{N}}(I)$ have the same $f$-vector, where ${\delta }_{\mathcal{F}}(I)$ (${\delta }_{\mathcal{N}}(I)$, respectively) is the facet (Stanley-Reisner, respectively) complex related to $I$. In this paper, we introduce the concepts of perfect set containing $k$ and perfect set without $k$. We study the $(n,d{)}^{th}$ perfect sets and show that $V(n,d)\ne \varnothing$ for $d\ge 2$ and $n\ge d+2$. Then we give some algorithms to construct $(n,d{)}^{th}$ $f$-ideals and show an upper bound for the $(n,d{)}^{th}$ perfect number.

12 pages