Periodic Parallelogram Polyominoes

TL;DR

This paper introduces periodic parallelogram polyominoes, extending bijections to compute their generating functions considering new statistics, and explores their classification into strips via a rotation operation.

Contribution

It extends existing bijections to include periodic parallelogram polyominoes and introduces a new statistic and classification method for these shapes.

Findings

Derived the generating function with respect to height, width, and intrinsic thickness.

Defined a rotation operation inducing an equivalence relation and computed the generating function of classes.

Connected the enumeration of strips to Pólya's theory for combinatorial enumeration.

Abstract

A periodic parallelogram polyomino is a parallelogram polyomino such that we glue the first and the last column. In this work we extend a bijection between ordered trees and parallelogram polyominoes in order to compute the generating function of periodic parallelogram polyominoes with respect to the height, the width and the intrinsic thickness, a new statistic unrelated to the existing statistics on parallelogram polyominoes. Moreover we define a rotation over periodic parallelogram polyominoes, which induces a partitioning in equivalent classes called strips. We also compute the generating function of strips using the theory of P\'olya.