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

William Keith william.keith at gmail.com
Mon Apr 11 10:04:48 CEST 2011

```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

```