# A code for square permutations and convex permutominoes

## TL;DR

This paper introduces a unified approach to enumerate square permutations and convex permutoninoes, providing new formulas and efficient algorithms for their random generation and analysis in convex grid configurations.

## Contribution

It presents a novel unified enumeration method for square permutations and convex permutoninoes, including refined formulas and linear-time random generation algorithms.

## Key findings

- Unified enumeration approach for both classes
- New refined formulas for generating functions
- Linear-time algorithms for random generation

## Abstract

In this article we consider square permutations, a natural subclass of permutations defined in terms of geometric conditions, that can also be described in terms of pattern avoiding permutations, and convex permutoninoes, a related subclass of polyominoes. While these two classes of objects arised independently in various contexts, they play a natural role in the description of certain random horizontally and vertically convex grid configurations. We propose a common approach to the enumeration of these two classes of objets that allows us to explain the known common form of their generating functions, and to derive new refined formulas and linear time random generation algorithms for these objects and the associated grid configurations.