The hexasticks

All polysticks of order 6 — figures built from 6 cells. Unit line segments of the square lattice joined end to end — also called polyedges or polyforms of the square grid's edges.

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• 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 A019988 (free)
• PASS · matches A151537 (onesided)
• PASS · matches A096267 (fixed)

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Free 222 polysticks

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 416 polysticks

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

Fixed 1,628 polysticks

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