Succession rules and Deco Polyominoes

In this paper, we examine the class of "deco" polyominoes and the succession rule describing their construction. These polyominoes are enumerated according to their directed height by factorial numbers. By changing some aspects of the "factorial" rule, we obtain some succession rules that describe various "deco" polyomino subclasses. By enumerating the subclasses according to their height and width, we find the following well-known numbers: Stirling numbers of the first and second kind, Narayana and odd index Fibonacci numbers. We wish to point out how the changes made on the original succession rule yield some new succession rules that produce transcendental, algebraic and rational generating functions.