Bar code for monomial ideals

The 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 p, by means of a bijection between these ideals and some integer partitions of p, which can be counted via determinantal formulas. This will be achieved by the Bar Code, a bidimensional diagram that allows to represent any finite set of terms M and desume many properties of the corresponding monomial ideal I, if M is an order ideal.