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%detconj.tex: A  Conjectured Explicit Determinant  Evaluation Whose Proof
%%a Plain TeX file by Douglas Hofstadter and Doron Zeilberger (2 pages)

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\centerline
{
\bf
A Conjectured Explicit Determinant Evaluation whose Proof
}
\centerline
{
\bf Would Make Us Happy  (and the OEIS Richer)
}

\centerline {By}
\medskip
\centerline
{\bf  Douglas R. Hofstadter and Doron Zeilberger }
\rm
\bigskip
{\bf Abstract}: We conjecture a certain explicit determinant 
evaluation, whose proof would imply the solution
of a certain enumeration problem that we have been working on, and 
that we find interesting.
We are pledging \$500 to the OEIS Foundation (in honor of the 
prover!) for a proof,
and \$50 (in honor of the disprover or his or her computer) for a 
disproof, as well as
(in the affirmative case only) a co-authorship in a good enumeration 
paper, that would immediately
a Hofstadter-number $1$, a Zeilberger-number $1$, 
an Erd\"os number $\leq 3$, an 
Einstein number $\leq 4$, and numerous other prestigious numbers.\halmos

In order to complete the proof of a certain enumeration problem that 
we have been working on
for the last few weeks, we need a proof of the following conjecture.

Let $d$ be a positive integer, and
let $M=M(d)$ be the following $2d \times 2d$ matrix with entries in 
$\{-1,0,1\}$.
For $1\leq a \leq 2d$ and $1 \leq b \leq d$,
$$
M_{a,2b-1}=
  \cases{
1 & if $a=2b$ ;\cr
  -1      &  if  $a=3b+1$;\cr
0 . & otherwise
}
$$
$$
M_{a,2b}=
  \cases{
1 & if $a=2b-1$ ;\cr
  -1      &  if  $a=b-1$;\cr
0 . & otherwise
}
$$

{\bf Conjecture}:  For every positive integer $d$, the following is true:
$$
\det M(d)=(-1)^d \quad .
$$


{\bf Comments}:

{\bf 1.} This conjecture came up in our current work in enumerative 
combinatorics.
Shalosh B. Ekhad kindly verified it for $d \leq 200$.
We have no idea how hard it is, and it is possibly not that hard, but right now
we are busy with other problems. We believe that the powerful and 
versatile techniques
of Krattenthaler[K1][K2] may be applicable, and possibly the
computer-assisted approach described in [Z] and already
nicely exploited in [KKZ] and [KT].

{\bf 2.} The Short Maple code in:
{\tt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/DetConj}
defines the matrix $M(d)$ (procedure {\tt M(d)}) and procedure
{\tt C(N)} verifies it empirically for all $d \leq N$.
So far {\tt C(200);} returned {\tt true}.

{\bf 3.} We are offering to donate \$500 to the OEIS Foundation for a 
proof and \$50 for a disproof,
with an explicit statement that the donation is in honor of the 
prover (or disprover).



{\bf References}


[K1] Christian Krattenthaler, {\it Advanced Determinant Calculus},
S\'em. Lothar. Comb. {\bf 42} (1999), B42q.
(``The Andrews Festschrift'', D. Foata and G.-N. Han
(eds.))
\hfill\break
{\tt http://www.mat.univie.ac.at/~kratt/artikel/detsurv.html}

[K2] Christian Krattenthaler, {\it Advanced Determinant Calculus: a 
complement},
Linear Algebra Appl. {\bf 411} (2005), 68-166
\hfill\break
{\tt http://www.mat.univie.ac.at/~kratt/artikel/detcomp.html}

[KKZ] Christoph Koutschan, Manuel Kauers, and Doron Zeilberger , {\it 
A Proof Of George Andrews' and David Robbins' q-TSPP Conjecture},
Proceedings of the National Academy of Science, {\bf 108\#6} (Feb. 8, 
2011), 2196-2199.

[KT] Christoph Koutschan and Thotsaporn Thanatipanonda,
{\it Advanced Computer Algebra for Determinants.},
Annals of Combinatorics, to appear.
\hfill\break
{\tt http://www.risc.jku.at/people/ckoutsch/det/}

[Z] Doron Zeilberger, {\it The HOLONOMIC ANSATZ II: Automatic 
DISCOVERY(!) and PROOF(!!) of Holonomic Determinant Evaluations},
Annals of Combinatorics {\bf 11} (2007), 241-247
\hfill\break
{\tt 
http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/ansatzII.html}


\bigskip
\hrule
\bigskip

Douglas R. Hofstadter, {\tt 
http://www.soic.indiana.edu/people/profiles/hofstadter-douglas.shtml}

Doron Zeilberger, {\tt http://www.math.rutgers.edu/\~{}zeilberg/}

{\bf First Version: Jan. 7, 2014}

{\bf This Version: April 17, 2014} (with first-named author added)

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