# Catalogue of Polyominoids

## Introduction

A *polyominoid*
is a polyform consisting of edge-connected squares in the polycube grid.
It is a 3-dimensional analogue of the polyomino.

Polyominoids may be generalized to other dimensions.
An *(n,k)-polyominoid*
is a set of *k*-dimensional
cells in the *n*-dimensional grid, connected at
(*k*−1)-dimensional cells.
Examples:

*n* | *k* | Name |

2 | 1 | polyline or polystick |

2 | 2 | polyomino |

3 | 1 | 3D polystick |

3 | 2 | ordinary polyominoid |

3 | 3 | polycube |

One can further generalize polyominoids by specifying the dimension of their
cell connections.
For example, a polyking may be regarded as a (2,2,0)-polyominoid.

Here I show all (3,2)-polyominoids with at most 4 cells.
Like polycubes, polyominoids may be *one-sided*
or *two-sided*.
One-sided means that distinct mirror images are counted as different
polyominoids.
Two-sided means that distinct mirror images are counted as the same
polyominoid.

## Enumeration

Cells
| Two-Sided
*A075679*
| One-Sided
*A056846* |

1 | 1 | 1 |

2 | 2 | 2 |

3 | 9 | 11 |

4 | 54 | 80 |

5 | 448 | 780 |

6 | 4650 | 8781 |

7 | 53611 | 104828 |

The diagrams below show the two-sided polyominoids.

## Monominoid

## Dominoids

## Trominoids

## Tetrominoids

Last revised 2022-05-31.

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Col. George Sicherman
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