Perfect Sets and $f$-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 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 $K[x_1,\dots ,x_n]$. We also consider the relation between an $f$-ideal and an unmixed monomial ideal.

15 pages

Cited by in corpus (1)

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