# [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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>

```