Fibonacci Polyominoes: Refined Enumeration and Dyck-Path Bijections
Abstract
We study Fibonacci polyominoes, a class of column-convex polyominoes whose lower boundary is a staircase path made of unit horizontal and vertical steps. We derive generating functions that enumerate these polyominoes by area and semiperimeter. The resulting formulas give refinements of the classical Fibonacci enumeration and lead to explicit expressions involving Catalan numbers. We give a bijection with labeled Dyck paths that provides combinatorial proofs of these formulas and translates natural statistics on Fibonacci polyominoes into peaks, returns, and related statistics on Dyck paths. This yields refinements by descents and left-to-right minima, including distributions governed by Narayana numbers. We also study consecutive columns of equal height, leading in the diagonal case to a refinement of the Catalan enumeration governed by Motzkin numbers.
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BibTeX
@misc{baril2026fibonacci,
title = {{Fibonacci Polyominoes: Refined Enumeration and Dyck-Path Bijections}},
author = {Jean-Luc Baril and José L. Ramírez and Samuel Ramírez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30097},
eprint = {2609.30097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30097},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fibonacci Polyominoes: Refined Enumeration and Dyck-Path Bijections
AU - Jean-Luc Baril
AU - José L. Ramírez
AU - Samuel Ramírez
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30097
UR - https://arxiv.org/abs/2609.30097
AB - We study Fibonacci polyominoes, a class of column-convex polyominoes whose lower boundary is a staircase path made of unit horizontal and vertical steps. We derive generating functions that enumerate these polyominoes by area and semiperimeter. The resulting formulas give refinements of the classical Fibonacci enumeration and lead to explicit expressions involving Catalan numbers. We give a bijection with labeled Dyck paths that provides combinatorial proofs of these formulas and translates natural statistics on Fibonacci polyominoes into peaks, returns, and related statistics on Dyck paths. This yields refinements by descents and left-to-right minima, including distributions governed by Narayana numbers. We also study consecutive columns of equal height, leading in the diagonal case to a refinement of the Catalan enumeration governed by Motzkin numbers.
ER -