[seqfan] Re: Counting polyominoes with peculiar symmetry

[Richard J. Mathar]

The refinements of the free polyominoes according
to symmetry are classified in Andrew Howroyd's sum formula
in A000105.
If we draw the diagonals of a n-omino (one diagonal per cell)
such that they are connected, they define a n-polystick of the square
lattice, A019988. I added links to drawings in A019988, and
a count by the order of the symmetry group of the n-polysticks in A348096.
[A348096 does not split the polysticks in 8 different symmetry groups
but only only in 4, i.e., without further refinement of the symmetry
element for groups of order 2 or 4.]