The Narrow-Minded and Ignorant Referee's Report [and Zeilberger's Response] of Zeilberger's Paper "Automaric CounTilings" that was rejected by Helene Barcelo and the Members of the Advisory Board [that includes(!) Enumeration Expert Mireille Bousquet-Melou] of the Journal of Combinatorial Theory-Series A.

>Report on a paper, "Automatic CounTiling," by D. Zeilberger.

>In this paper the author describes a program
>that computes the rational generating function for the tilings of rectangles
>of a given fixed height k, and of varying length n. A finite list of
>usable tiles is given as input, and the program builds on "its own"
>a system of linear equations of which one of the component of the solution
>is the required generating function.

>While it is very useful and quite difficult to have a program that can enumerate
>tilings, as the author has done, this manuscript is a rather sketchy description of a
>computer program and doesn't seem to contain any original mathematics. 

Mathematics my foot! Algorithms are mathematics too, and often more
interesting and definitely more useful. My Maple program can discover and prove
completely automatically for an ARBITRARY set of tiles and ARBITRARY width.
In principle it contains many JCT-A-style papers.

Programming is much much harder than doing mathematics.
I bet that the referee will never ever be able to write such a program
(if he or she did, they would be able to appreciate the novelty)

>In particular, the section on "almost" automatic proof of Kasteleyn's theorem 
>is too sketchy and not clear to me. 

It only shows  that you are not qualified to referee this paper. It was very
clear.

>I am not sure what "almost" precisely
>means. 

"almost" meanst that the computer finds a completely rigorous (and automatic!) proof of 
Kasteleyn's theorem for any fixed m but arbitrary n!

>Moreover, as a minor point, I am puzzled by the statement on page 6 that "it is
>readily seen that W_{m,n}(z,z') is expressible as a product of m/2
>different dilations of U_n(z)." For instance, W_{4,6} is
>irreducible.

It is irreducible over the rationals. The dilations U_n(beta z)
are with beta algebraic. This is obvious if one looks at the explicit
product featuring in the formula that involves members of a cyclotomic field.


> Also, why does m have to be fixed? By symmetry W_{m,n}
>should also be expressible as a product of n/2 (instead of m/2)
>different dilations of U_m(z) (instead of U_n(z)). 

Of course, but what's your point? You can fix either m OR n.

>An analysis of the
>generating function for dimer tilings of a rectangle of fixed width
>appears in a paper by Stanley, [1].

It is very nice, but completely human-made.

>In conclusion, although a good paper of this kind would be interesting,
>I dont believe that the present manuscript meets the required standards for 
>acceptance in the Journal.


You are right here. It is way too good for it! I kick myself for being nice and
submitting this paper to your narrow-minded pretentious journal! It won't happen again.


   -Doron Zeilberger

Opinion 61 of Doron Zeilberger