[seqfan] Re: n x n matrices with adjacent entries differing by +/- 1

[Marc LeBrun]

Edwin, if you'll forgive a less-than rigorous 
argument (ie it may be just the late coffee 
talking) I think I can see how it's not so 
surprising that these might be 
isomorphic.  Consider constructing a trace of the 
work done by coloring algorithms that step 
sequentially through the cells in some order.  In 
your up/down version the "next" cell can only be 
one of two colors, and in A068253's mod-3 version 
the choice is also limited to two possible 
outputs, although the kind of "paint" actually 
output differs.  I imagine you can probably do 
something like take the deltas in your version, 
and cumulatively sum them mod-3 to get A068253's patterns, or the like...?

Anyway, regardless, it might also be interesting 
to see what sequences you get when the constraint 
wraps around the edges (with or without a reflection)?


At 06:54 PM 11/1/2008, you wrote:
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>On Mon, 27 Oct 2008, Benoît Jubin wrote:
>>
>>It would also be interesting to consider the entries of the matrix in
>>Z/kZ (that is, 1 and k would also differ by 1). And also the same
>>sequence for entries in Z or N, the upper-left term being 0.
>
>I computed for n = 1 to 8 the number of n x n 
>matrices with entries in Z, the upper left entry 
>= 0, and adjacent entries (in the same row or 
>column) differing by +/- 1. I got the following:
>
>1, 6, 82, 2604, 193662, 33865632, 13956665236, 13574876544396
>
>This  matches the first 8 terms of
>
>  http://www.research.att.com/~njas/sequences/A068253
>
>which is defined as:
>
>  1/3 of the number of colorings of an n X n square array with 3 colors
>
>It is too much to imagine the sequences are 
>different, yet I also cannot imagine they 
>coincide by looking at the definitions. Does anyone see a connection?
>
>--Edwin
>
>
>
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