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

[Ron Hardin]

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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