Perfect Sets and $f$-Ideals

TL;DR

This paper introduces perfect subsets to characterize $f$-ideals in square-free monomial ideals, providing formulas for counting degree 2 $f$-ideals and exploring their relation to unmixed ideals.

Contribution

It defines perfect subsets of $2^{[n]}$, characterizes $f$-ideals of degree $d$, and derives a counting formula for degree 2 $f$-ideals using graph-set correspondences.

Findings

Characterization of $f$-ideals via perfect subsets.

A decomposition of $V(n, 2)$ for counting degree 2 $f$-ideals.

Relation established between $f$-ideals and unmixed monomial 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 and study perfect subsets of $2^{[n]}$ and use them to characterize the $f$-ideals of degree $d$. We give a decomposition of $V(n, 2)$ by taking advantage of a correspondence between graphs and sets of square-free monomials of degree $2$, and then give a formula for counting the number of $f$-ideals of degree $2$, where $V(n, 2)$ is the set of $f$-ideals of degree 2 in $K[x_1,\ldots,x_n]$. We also consider the relation between an $f$-ideal and an unmixed monomial ideal.