[seqfan] Re: Domino Tiling ^ 2 = city block distance one permutation?

[William Keith]

On Mon, Apr 11, 2011 at 10:27 AM, Ron Hardin <rhhardin at att.net> wrote:

> My difficulty is that that orients all the cycles at once, when it seems to
> me
> that the cycles have to take on every possible orientation independently.
>
> There's 2 orientations depending on whether youtake AB or BA, whereas you
> need
> 2^number of cycles orientations.
>
> Obviously my idea is wrong by the counts, but I don't see how.
>
>  rhhardin at mindspring.com
> rhhardin at att.net (either)
>

For each collection of cycles, you can make each choice independently.  If
there are n cycles, order the cycles by uppermost leftmost element, and
start with the A,B choice in each that assigns them all the counterclockwise
orientation.  That's a pair of tilings.  Call it 00..00.  Then take the last
cycle and reverse the choice of pair.  Call that 00..01.  Then put it back,
reverse the choice of pair in the next to lastmost, call that 00..10.  Now
keep that and also reverse the last.  Call that 00..11.  Etc. until 11..11,
which has all the pirs reversed.  Each corresponds to a choice of tiling
pairs.

William Keith