The Structure of Local Rings with Singleton Basis and Their Enumeration

A local ring is an associative with unique maximal ideal. We associate each Artinian singleton basis four invariants (positive integers) p,n,s,t. The purpose of this article to describe the structure such rings and classify them (up isomorphism) same invariants. Every can be constructed over its coefficient subring by a certain polynomial called associated polynomial. These polynomials play significant role in enumeration.