Polyominoes by Directedness

This table categorizes the polyominoes up to octominoes by their directedness class. For classes that have no members with size 8 or less, representatives of the smallest possible size are presented.

Sections stacked on top of each other have a hierarchial relationship: for example, all stack polyominoes are bar charts. Additionally, polyominoes with equivalent boundary words are grouped together.

Rectangle ( $\textrm{Dir}_{4}^{4}$ )

ruld

Wedge ( $\textrm{Dir}_{4}^{3}$ )

ru(ru)*ld

ru²ld

ru³ld

Staircase ( $\textrm{Dir}_{4}^{2\textrm{t}}$ )

ru(ru)*ld(ld)*

ru²ld²

ru³ld²

ru³ld³

ru⁴ld³

ru⁴ld²

ru⁴ld⁴

Stack ( $\textrm{Dir}_{4}^{2\textrm{c}}$ )

ru(ru)*(lu)*ld

ru²luld

ru³luld

Fork ( $\textrm{Dir}_{4}^{1}$ )

ru(ru)*(lu)*ld(ld)*

ru²luld²

ru³luld²

ru³luld³

ru²lu²ld²

ru⁴luld³

Bar chart ( $\textrm{Dir}_{3}^{2\textrm{c}}$ )

ru(ru|lu)*ld

ru²luruld

ruluruld

ru²lurululd

ruluruluruld

ruluru²ld

Diamond ( $\textrm{Dir}_{4}^{0}$ )

ru(ru)*(lu)*ld(ld)*(rd)*

ru²luld²rd

ru³luld²rd

ru³luld³rd

Wing ( $\textrm{Dir}_{3}^{1}$ )

ru(ru|lu)*ld(ld)*

ruluruld²

ru²luruld²

rulurululd²

ruluruld³

ruluru²ld²

ru²lurululd²

ru²luruld³

ruluru²ld³

ru³luruld²

rulurululd³

ru²luru²ld²

ruluru²luld²

Crescent ( $\textrm{Dir}_{3}^{0}$ )

ru(ru|lu)*ld(ld)*(rd)*

ruluruld²rd

ru²luruld²rd

ruluruld²rd²

ru²lurululd²rd

ru²luruld³rd

ruluru²ld²rd

ru²luruld²rd²

ruluruld³rd²

ruluru²ld³rd

Antler ( $\textrm{Dir}_{2\textrm{c}}^{1}$ )

ru(ru|lu|ld(ld)*lu)*ld(ld)*

ruldluruld

ru²ldluruld

rululdluruld

ruluruldluld

Range chart ( $\textrm{Dir}_{2\textrm{t}}^{0}$ )

ru(ru|lu)*ld(ld|rd)*

ruluruldrdld

ru²luruldrdld

ru²luruld²rdld

ru²luruldrdldrd

Bent tree ( $\textrm{Dir}_{2\textrm{c}}^{0}$ )

ru(ru|lu|ld(ld)*lu)ld(ld)*(rd)*

ruldluruldrd

ru²ldluruldrd

ruluruldluldrd

Tree ( $\textrm{Dir}_{1}^{0}$ )

ru(ru|lu|ld(ld|rd)*lu)*ld(ld|rd)*

ruluruldrdluld

ruluruldluldrdld

Other ( $\textrm{Dir}_{0}^{0}$ )

ruldluruldrurdld

rurdruluruldluldrdld

rurdruluruldrdluldrd

rurdru²luruldrdluld

rurdruluruldrdluld

rurdru²ldluruldrdld

Polyominoes by Directedness

Rectangle (Dir44\textrm{Dir}_{4}^{4}Dir44​)

Wedge (Dir43\textrm{Dir}_{4}^{3}Dir43​)

Staircase (Dir42t\textrm{Dir}_{4}^{2\textrm{t}}Dir42t​)

Stack (Dir42c\textrm{Dir}_{4}^{2\textrm{c}}Dir42c​)

Fork (Dir41\textrm{Dir}_{4}^{1}Dir41​)

Bar chart (Dir32c\textrm{Dir}_{3}^{2\textrm{c}}Dir32c​)

Diamond (Dir40\textrm{Dir}_{4}^{0}Dir40​)

Wing (Dir31\textrm{Dir}_{3}^{1}Dir31​)

Crescent (Dir30\textrm{Dir}_{3}^{0}Dir30​)

Antler (Dir2c1\textrm{Dir}_{2\textrm{c}}^{1}Dir2c1​)

Range chart (Dir2t0\textrm{Dir}_{2\textrm{t}}^{0}Dir2t0​)

Bent tree (Dir2c0\textrm{Dir}_{2\textrm{c}}^{0}Dir2c0​)

Tree (Dir10\textrm{Dir}_{1}^{0}Dir10​)

Other (Dir00\textrm{Dir}_{0}^{0}Dir00​)