Hilbert-Samuel sequences of homogeneous finite type

This paper deals with the problem of the classification of the local graded Artinian quotients K[x,y]/I where K is an algebraically closed field of characteristic 0. They have a natural invariant called Hilbert-Samuel sequence. We say that a Hilbert-Samuel sequence is of homogeneous finite type, if it is the Hilbert-Samuel sequence of a finite number of isomorphism classes of graded local algebras. We give the list of all the Hilbert-Samuel sequences of homogeneous finite type in the case of algebras generated by 2 elements of degree 1.