Centered Ascending Polyominoes

Polyomino enumeration remains a central and challenging problem in enumerative combinatorics. Among convex polyominoes, geometrically defined subclasses such as L-convex and Z-convex families exhibit markedly different enumerative behaviours, showing how geometric constraints influence the nature of the generating function. In this paper, we introduce a new class of convex polyominoes, called centered ascending polyominoes, which extends L-convex polyominoes and forms a proper subclass of the Z-convex ones. We provide their first exact enumeration with respect to the semiperimeter (size). Our approach is based on a geometric characterisation involving horizontal inclusion and north–east shift constraints between rows, leading to a decomposition into six natural subclasses. We construct a generating tree that uniquely produces objects of size n + 1 from those of size n. This yields a system of functional equations whose solution provides an explicit algebraic generating function, a closed counting formula, and the corresponding asymptotic estimate.