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%and Margie Readdy, About Your Two Conjectures,
%But One is FAMOUS, While The Other Is FALSE'
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\bf
\centerline
{
I Am Sorry, Richard Ehrenborg and Margie Readdy, About Your Two Conjectures,
}
\centerline
{
But One is FAMOUS, While The Other Is FALSE
}

  
\rm
\bigskip
\centerline{ {\it 
Doron ZEILBERGER
}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg@math.rutgers.edu} \hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/   .}
First version: July 2, 2002.  
Accompanied by the Maple package {\tt EHRENBORG} available from
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/programs.html} .
Supported in part by the NSF. 
Exclusively published in the `Personal Journal of Ekhad and Zeilberger'
\hfill\break
(url: {\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/pj.html}.)
} 
}
 
R. Ehrenborg and M. Readdy[ER] raised two conjecture. The first
one was that a certain triple-sum indexed by permutations
of length $n$ equals $1!3!5! \cdots (2n-1)!$. I don't remember
the details right now, but I immediately recognized it
as equivalent to a special case of the celebrated Mehta Integral
Identity
(that was ``open'' between 1965 and 1980, until one time Freeman Dyson
mentioned it to Atle Selberg, and the latter immediately deduced it 
from his (then obscure, but now famous) 1944 integral).

The second conjecture was more interesting.
It claimed that there are
exactly $k!$
Down-Up {\it involutions} 
of length $2k$. Richard checked it for
$k=1, \dots, 5$. Alas, using my Maple package
{\tt EHRENBORG}, it turns out
(type {\tt nDUI(12);}), that there are only
$716$ Down-Up involutions of length $12$, which is $4$ short of
$6!$.

For the sake of completeness, here are the
first $24$ terms of the enumerating sequence
for Down-Up involutions:

{\tt
[1, 1, 1, 2, 3, 6, 11, 24, 51, 120, 283, 716, 1833, 4948, 13561, 38788,
    112745, 339676, 1039929, 3283876, 10532747, 34717276, 116158851, 398257012]
}.


{\bf REFERENCE}

[ER] Richard Ehrenborg and Margie Readdy, 
{\it private communication}, during lunch
with Theodore Ehrenborg,
at the Indian restaurant {\bf Masala Grill}, Princeton, NJ,
June 11, 2002.

\bye





