The pentiamonds

All polyiamonds of order 5 — figures built from 5 cells. Edge-joined equilateral triangles of the triangular lattice, alternating point-up and point-down.

✓ 15/15 verification checks passed

• PASS · all forms have n cells
• PASS · all forms connected
• PASS · fixed forms distinct
• PASS · orbit count: free -> fixed
• PASS · orbit count: free -> one-sided
• PASS · orbit count: one-sided -> fixed
• PASS · free <= one-sided <= fixed
• PASS · canonical form idempotent
• PASS · canonical form group-invariant
• PASS · encoding round-trips
• PASS · free forms expand to the fixed set
• PASS · free forms expand to the one-sided set
• PASS · matches A000577 (free)
• PASS · matches A006534 (onesided)
• PASS · matches A001420 (fixed)

drawing…

Free 4 polyiamonds

Two figures are the same free form if one can be turned into the other by translation, rotation or reflection. This is the usual count — the one that gives 12 pentominoes.

One-Sided 6 polyiamonds

One-sided forms may be translated and rotated but not flipped over, so a chiral figure and its mirror image are counted separately.

Fixed 36 polyiamonds

Fixed forms may only be translated. Every distinct rotation and reflection is counted as its own figure.