Fibonacci and Catalan Numbers Meet in Staircase Polyominoes
This paper derives multivariate generating functions to refine the enumeration of Fibonacci polyominoes.
This paper introduces a new approach to enumerating Fibonacci polyominoes by tracking additional perimeter and area parameters.
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Abstract
More Like ThisWe study Fibonacci (staircase) polyominoes, a class of column-convex polyominoes whose lower boundary is a staircase with unit vertical steps. We derive multivariate generating functions that refine Turban's Fibonacci-number enumeration by tracking additional perimeter and area parameters. The proofs use a catalytic functional equation and, in a perimeter specialization, the kernel method, leading to explicit closed forms and Catalan-number coefficient formulas.