[seqfan] Re: Unique Products Regarding Binary Matrices -- really arrays

[Marc LeBrun]

>="hv at crypt.org" <hv at crypt.org>
> I'm not at all sure what you're asking for here, but for each interpretation
> I can imagine:
> 
> Let A be any such matrix; define B: B_{i,j} = 1 - A_{i,j}.
> 
> Then I think each of the 2n products for B is the same as the corresponding
> product for A, so A is not unique.

This may seem a fine point of terminology, but I recommend distinguishing
between the terms ARRAY and MATRIX, reserving the latter ONLY for objects
where operations such as matrix addition, multiplication, determinant,
permanent, etc are relevant.  Arrays are simpler, more abstract, objects
(what might be called a "super class" in CS lingo).

It's natural to casually mix up "array" and "matrix" in informal queries and
eMail -- such as with the interesting sequence here -- but it's imprecise
and may be misleading when "matrix" drags along all these extra concepts.

For instance "matrix" typically implies the elements are drawn from a ring.
But in this "binary" example, which one?  Does 1+1=0 or 1+1=1?  Actually of
course it's totally irrelevant for the combinatorial sequence here.

The topic at hand isn't really defined in terms of "binary matrices" at all,
but rather by counting patterns in square arrays composed from an alphabet
with two symbols.

With that interpretation changing the label alphabet from <0,1> into <1,0>,
or even <X,Y> leaves the count invariant.

As a bonus the more generic setting can suggest interesting generalizations:

  *  Alphabets with more than 2 symbols (Say N symbols? Or even N^2?)

  *  Rectangular MxN arrays (giving M+N products, and an OEIS table)

  *  Edgeless arrays with toroidal or moebius wrap-around connectivity

  *  Higher-dimensional arrays (cubes giving N^3 products, etc)

  *  Other symmetries (say triangular or hexagonal, giving 3N products)