# Bar code for monomial ideals

## TL;DR

This paper introduces the Bar Code, a bidimensional structure that represents finite sets of monomials, enabling the counting of 0-dimensional stable and strongly stable ideals in two and three variables based on their Hilbert polynomial.

## Contribution

The paper defines the Bar Code structure and establishes a connection between stable monomial ideals and integer partitions for enumeration purposes.

## Key findings

- Counted stable and strongly stable ideals in 2 and 3 variables.
- Connected monomial ideals to integer partitions using the Bar Code.
- Provided formulas for counting ideals based on Hilbert polynomials.

## Abstract

Aim of this paper is to count $0$-dimensional stable and strongly stable ideals in $2$ and $3$ variables, given their (constant) affine Hilbert polynomial. To do so, we define the Bar Code, a bidimensional structure representing any finite set of terms $M$ and allowing to desume many properties of the corresponding monomial ideal $I$, if $M$ is an order ideal. Then, we use it to give a connection between (strongly) stable monomial ideals and integer partitions, thus allowing to count them via known determinantal formulas.