Enumerations and Bijections for Stanley Polyominoes

TL;DR

This paper derives generating functions for Stanley polyominoes and establishes bijections with various combinatorial structures, including solving an open problem relating parallelogram polyominoes and coin fountains.

Contribution

It introduces new generating functions for Stanley polyominoes and provides novel bijections with Dyck paths, Ferrer diagrams, and Motzkin paths, including solving an open problem.

Findings

Derived explicit generating functions for Stanley polyominoes.

Established bijections with Dyck paths, Ferrer diagrams, and Motzkin paths.

Solved the open problem connecting parallelogram polyominoes and coin fountains.

Abstract

Stanley polyominoes are a subclass of parallelogram polyominoes in which each row begins strictly to the right of the beginning of the previous row and ends strictly to the right of the end of the previous row. In this paper, we derive generating functions for Stanley polyominoes based on the numbers of columns and rows, area, semiperimeter, and numbers of interior points and edges. We also establish combinatorial connections through bijections with other combinatorial structures such as Dyck paths, skew Ferrer diagrams, and peakless Motzkin paths. As a byproduct, we answer the open question of finding a bijection between parallelogram polyominoes of area $n$ and coin fountains with $n$ coins in the even-numbered rows and $n-k$ coins in the odd-numbered rows.