Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO.

This paper studies a class of polyominoes. The new class is defined through a representation on the James abacus through nested chain which denoted Ω -nested Abacus. Depended on partition, beta number, nested chain we defined the structural condition this class. A local transformations, called SSPT-transformation and MSPT-transformation are formulated on the even chains. Then, we drive explicit formulas for the number of position in each chain, number of even chain and the total number of all chains. The structure of new class lead to enumeration formulas for the objects produced after application new transformation. To enhance further, based on these classes, generating functions are also being formulated by employing enumeration of combinatorial objects (ECO). In ECO method, each object is obtained from smaller object by making some local expansions. These local expansions are described in a simple way by a succession rule which can be translated into a function equation for the generating function.