Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO

This paper studies a new family of polyominoes. The family is defined through a graphic representation called James diagram, also known as the nested chain abacus. This form is referred to as the Ω-nested Abacus. The proposed class is defined by geometric constraints satisfied by sets of rows, columns, and chains. Through the application of combinatorial approaches and variable generating function, we enumeration of this new class of polyominoes. During the new construction process transformation is formulates which plays a crucial role in the enumerate of a family of classes. To further extend this study generating function for theses class are formulated using an enumeration method known as the Enumeration Combinatorial Object (ECO) technique. ECO method, derived from a smaller one though a series of local expansion. Theses expansion are systematically defined by succession rule which can subsequently be expressed as a functional equation for the generating function.