POLYOMINO TILINGS

Polyomino Count

Polyomino count by type.

Formulas

$P_{\text{free}}(n)$ - number of free $n$-ominoes

$P_{\text{holeless}}(n)$ - number of free $n$-ominoes without holes

$P_{\text{holey}}(n)$ - number of free $n$-ominoes with holes

$P_{\text{one-sided}}(n)$ - number of one-sided $n$-ominoes

$P_{\text{fixed}}(n)$ - number of fixed $n$-ominoes

$P_{\text{none}}(n)$ - number of $n$-ominoes without symmetry

$P_{\text{mirror 90°}}(n)$ - number of $n$-ominoes with one axis of reflection symmetry at 90° to the gridlines

$P_{\text{mirror 45°}}(n)$ - number of $n$-ominoes with one axis of reflection symmetry at 45° to the gridlines

$P_{\text{C2}}(n)$ - number of $n$-ominoes with rotational symmetry of order 2

$P_{\text{D2 90°}}(n)$ - number of $n$-ominoes with with two axis of reflection symmetry at 90° to the gridlines

$P_{\text{D2 45°}}(n)$ - number of $n$-ominoes with with two axis of reflection symmetry at 45° to the gridlines

$P_{\text{C4}}(n)$ - number of $n$-ominoes with rotational symmetry of order 4

$P_{\text{D4}}(n)$ - number of $n$-ominoes with full symmetry of square

$P_{\text{holeless}}(n) = P_{\text{free}}(n) - P_{\text{holey}}(n)$

$P_{\text{free}}(n) = P_{\text{none}}(n) + P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + P_{\text{C2}}(n) + P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + P_{\text{C4}}(n) + P_{\text{D4}}(n)$

$P_{\text{one-sided}}(n) = 2P_{\text{none}}(n) + P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + 2P_{\text{C2}}(n) + P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + 2P_{\text{C4}}(n) + P_{\text{D4}}(n)$

$P_{\text{fixed}}(n) = 8P_{\text{none}}(n) + 4(P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + P_{\text{C2}}(n)) + 2(P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + P_{\text{C4}}(n)) + P_{\text{D4}}(n)$

Attributions

1. Free polyominoes have been counted by Toshihiro Shirakawa (http://oeis.org/A000105).
2. One-sided polyominoes have been counted by Toshihiro Shirakawa (http://oeis.org/A000988).
3. Free polyominoes with holes and polyominoes of each symmetry have been counted by Tomas Oliveira e Silva (http://sweet.ua.pt/tos/animals/a44.html).
4. Fixed polyominoes have been counted by Iwan Jensen (http://www.ms.unimelb.edu.au/~iwan/animals/Animals_ser.html).