[seqfan] Re: Domino Tiling ^ 2 = city block distance one permutation?
Ron Hardin
rhhardin at att.net
Mon Apr 11 11:27:34 CEST 2011
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)
----- 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 4:04:48 AM
> Subject: [seqfan] Re: Domino Tiling ^ 2 = city block distance one permutation?
>
> which is
>
> > http://oeis.org/A099390/table the number of domino tilings of a nXk grid
> >
> > So: take two tilings A and B as defining a permutation. But how?
> >
> > Obviously where they align, it's a pure swap. Where they cross, they
> > define
> > cycles.
> >
>
> Nice way to do it.
>
>
> > But I don't see how they define a direction to each cycle. Any ideas?
> > There
> > are many disjoint cycles in general, and each one has to get a specific
> > orientation.
> >
>
> The pairs of tilings are ordered. Take the uppermost leftmost element in
> each cycle in whatever ordering of the entries you want, and it will be on
> one side or the other of the domino on the A side. Go along that direction
> for that orientation of the cycle. The other direction arises because you
> can switch the A and B lists, which will send that element in the opposite
> direction along the same cycle.
>
> William Keith
>
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