[seqfan] Re: n x n matrices with adjacent entries differing by +/- 1
Benoît Jubin
benoit.jubin at gmail.com
Sun Nov 2 04:14:18 CET 2008
They are actually the same: let us call the three colors 0,1,2, and
suppose the upper-left color is 0 (the 1/3 in the definition).
There is a bijection between the set of matrices with entries in Z,
upper-left entry 0 and adjacent entries differing by +/-1, and the set
of 3-colored matrices with upper-left being 0: it is simply "modulo
3". Indeed, the constraints correspond under this function between
sets of matrices: for the coloring, the constraint is that the
difference between adjacent entries is +/-1 in Z/3Z, and injectivity
comes from the fact that 3>2.
Benoit
2008/11/1 Edwin Clark <eclark at math.usf.edu>:
> 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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