Fibonacci and Catalan Numbers Meet in Staircase Polyominoes
This study addresses the refined enumeration of column-convex polyominoes whose lower boundary consists of unit vertical steps forming a staircase. By constructing a multivariate generating function that simultaneously tracks area and perimeter, and by employing catalytic functional equations together with the kernel method, the authors derive—for the first time—a closed-form solution. This work not only refines Turban’s classical Fibonacci-based counting result but also reveals, for the first time, an explicit appearance of Catalan numbers in the distribution of certain perimeters. Furthermore, it provides an exact enumeration formula that jointly accounts for both area and perimeter statistics.
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