[seqfan] Re: n x n matrices with adjacent entries differing by +/- 1
Marc LeBrun
mlb at well.com
Sun Nov 2 04:22:27 CET 2008
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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