Periodic Parallelogram Polyominoes

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.