[seqfan] Re: Domino Tiling ^ 2 = city block distance one permutation?
Ron Hardin
rhhardin at att.net
Mon Apr 11 13:35:19 CEST 2011
So cycles don't define a unique choice of an A B pair, but you can swap the
cycle portions of an A B pair to get other tiling pairs with the same cycles, to
cover each orientation of each cycle.
rhhardin at mindspring.com
rhhardin at att.net (either)
----- Original Message ----
> From: William Keith <william.keith at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Mon, April 11, 2011 7:16:02 AM
> Subject: [seqfan] Re: Domino Tiling ^ 2 = city block distance one permutation?
>
> 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
>
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