[seqfan] Counting polyominoes with peculiar symmetry

[Allan Wechsler]

Take a polyomino and mark each of its square cells by drawing one of its
two diagonals. But once you choose the first diagonal, all the others are
forced because I require these diagonal markings to meet at their
endpoints. (That is, /\ is fine, but // isn't allowed.)

Depending on your choice of the first diagonal, you can mark up any
polyomino in two ways, but sometimes these two ways turn out to be
isomorphic. That only happens for polyominoes that have some symmetry, but
it doesn't happen for all symmetric polyominoes, just some of them.

By my hand count, the number of qualifying polyominoes of orders 1 through
6 are:

1,1,1,3,4,10

But I am not sure of the hexomino number. None of the 11 matches at OEIS
say anything about polyominoes.

I hope some polyomino counters out there understand what I mean, and can
verify or contradict my counts, and tell me whether the resulting sequence
is already in OEIS or not.