There are two triominoes. One is simple a 1x3 rectangle and the other is a small
L-shaped piece.
Using any combination of these two triominoes and in any orientation, how many ways can a 3 by N rectangle be tiled?
Rotations and reflections of an asymmetrical solution are to be considered distinct.
(In reply to
re: thanks by Brian Smith)
I put my translation of the Germain Kreweras paper here.
In it you will see a proof of completeness of the list of 5 allowed tiling
forms, a proof that is similar to your own where the use of the
L-shaped triominis is confined to the very ends of the 3 x N area.
This again is because using the L-shape within the area would lead
to closed regions yet to tiled are not multiples of 3.
You can also see his usage of generating functions that stumped me.
At the end the application to gerrymandering is made. From that
simplistic model he shows that it is not at all difficult to produce
very contrived results, if the drafter of districts is so inclined
(as we know all too well).
Original French Kreweras paper is here and here.
Edited on November 12, 2023, 11:07 am
re(2): thanks
