By Shalosh B. Ekhad, Vince Vatter and Doron Zeilberger
Written: Sept. 14, 2005.
Last Update of this page: Oct. 1, 2006.
Note added Oct. 1, 2006: We just found out that this masterpiece was
narrow-mindedly rejected by Electronic Research Announcements of the American
Mathematical Society. Since editing and refereeing is still done by humans,
we decided to forgoe publishing in a "real" journal (even an electronic one),
and keep this in our respective websites.
You may find it instructive and/or amusing to see
the referee's stupid report and our response to it.
Some conjectures are irresistable. Nick Loehr and Greg Warrington told
Bruce Sagan, who told Vince Vatter, who told me, that they
believe that a certain easy-to-describe set of lattice paths
are enumerated by 10**n. Such a simple result must be trivial to prove.
And sure it is! So what if it took Vince and I a week
to write a Maple program, another two weeks to debug it completely,
and Shalosh 5 seconds to find a `grammar' in terms of a family (binary) tree
with 81 leaves and 80 internal vertices, and another 30 seconds to
prove it rigorously. It sure is trivial, at least a posteriori.
Inspired by our proof, Nick and Greg and Bruce Sagan found
a `computer-free' proof (for what it is worth), but,
more interestingly, they used their humanized approach
to generalize it. Stand by for their forthcoming paper.
[Once it is ready we will put a link to it here.]
Added Oct. 23, 2005: I loved Jonas's proof so much that I
wrote an
expository article
about it in my Personal Journal.
.pdf
.ps
.tex
Exclusively published in the
Personal Journal of Ekhad and Zeilberger and in
Vince Vatter's website .
Added Sept. 23, 2005: These humans, with all their flaws,
are sometimes surprisingly ingenious. Exactly a week after
our article was posted at arXiv.org, the clever
human Jonas Sjöstrand found a
beautiful bijecive proof of the most general conjecture,
thereby even beating Loehr, Sagan, and Warrington.
Congratulations, Jonas.
Of course, as we mention in remark TEN of our paper, the main
point of our paper was to develop and illustrate a methodology
of completely computer-generated research, and this methodology
should extend to many other cases where the answers are not
so beautiful, and hence it is very unlikely that there exist
nice human proofs. At any rate we greatly admire Jonas's proof.
Important: This article is accompanied by Maple
package
TEN
that did everything.
Added Sept. 20, 2005: TEN now has a very verbose version,
TENverbose, that gives all the details, and makes
the proof completely humanly-readable.
The
input, would yield the very long-winded
output.
Doron Zeilberger's List of Papers