Fibonacci and Catalan Numbers Meet in Staircase Polyominoes

2026-06-11

AI Summary

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.

Abstract

We 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.

Problem

• Fibonacci numbers
• Catalan numbers
• staircase polyominoes
• column-convex polyominoes
• perimeter and area enumeration

Innovation

• staircase polyominoes
• multivariate generating functions
• catalytic functional equation
• kernel method
• Catalan numbers