Density of $f$-ideals and $f$-ideals in mixed small degrees

TL;DR

This paper studies the density and construction of $f$-ideals, a special class of squarefree monomial ideals, showing their density diminishes with more variables and providing algorithms for small degrees.

Contribution

It establishes the asymptotic density of $f$-ideals generated in fixed degrees and introduces new algorithms for constructing $f$-ideals in small degrees.

Findings

$f$-ideals generated in fixed degrees have density zero as variables increase

New algorithms for constructing $f$-ideals in small degrees

Analysis of the relationship between Stanley-Reisner and facet complexes

Abstract

A squarefree monomial ideal is called an $f$-ideal if its Stanley-Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct $f$-ideals generated in small degrees.