
   \magnification=1200
   \hsize=6.5truein
   \vsize=9truein
   \footline={\hfil}
   \headline={\ifnum\pageno=1 \hfil \else \tenrm\hfil\folio\fi}
   \font\SC=cmcsc10 scaled 1440
   \font\sc=cmcsc10 at 10truept
   \def\uline{\hbox to.7em{\hfil\hbox to.5em{\hrulefill}\hfil}}
   \line{\SC Rutgers University \hfil\sc (New Brunswick) Piscataway
   New Jersey 08854-8019}
   \vskip-0.75em\line{\hrulefill}\vskip-0.25em
   \line{\sc Department of Mathematics-Hill Center
        \hfil     e-mail: {\tt zeilberg@math.rutgers.edu}}
\line{(732) 445--1326 (voice) \hfil (732) 445--5530 (Fax)}

   %
   % If you prefer non-proportional spacing and a typewriter font,
%include this:
   %
   %\tt\raggedright
   %here is a "spring" for short letters
   \vfill
   \bigskip
   \line{\hskip 0.5\hsize
%       Date goes here ***********************************
Oct. 17, 2005 \hfil}
   \bigskip
% address here
{\obeylines
NSF Postdoctoral Research Fellowship
National Science Foundation
Arlington, VA
}
   \bigskip

   \medskip\noindent
% Salutation here:  Dear ...
Dear Committee,
   \smallskip
   {\openup 1.5\jot

% Body of letter here.
This is a letter of recommendation on behalf of my doctoral student
Vince VATTER
who is applying for an NSF Postdoctoral Research Fellowship
(MSPRF).

I already knew of Vince's work before he came to Rutgers.
My good friend Bruce Sagan told
me to watch out for his brilliant student, who did
undergraduate research under him and already had one accepted
article and two in preparation, a record that would
most likely suffice for a Ph.D.! I had a chance to
review that first paper in my capacity as Editor of
Adv. Appl. Math. This paper is a really significant
contribution to permutation enumeration.
Their work is now covered in detail in Mikl\'os B\'ona's recent
book ``Combinatorics of Permutations,'' the definitive
monograph in the field.

Vince is currently a fourth-year Ph.D. candidate,
and although he has yet to complete all the formalities
to graduate (his defense is scheduled for November, 2005), he
has begun a one-year postdoc at the University
of St.~Andrews, where he continues to do first-class
research. I can safely
say that he is the best Ph.D. student that I have ever guided
(so far I have graduated twelve students, and have five current
students). I have also interacted with many graduate students in 
my many courses, and would definitely rank him in the top
5 percent of all students I have known, if not higher.

Vince is almost unique in combining deep theoretical insight
with uncanny computer-algebra wizardry.  That will make him
ideally suited to pursue his proposed research
with Richard Stanley.

Vince's research output is truly amazing, in {\it both}
quantity and quality. So far he has ten finished papers
(seven published or accepted,
and three under consideration).

Vince has shown considerable breadth in his research already.  He has
written on graph theory (``Maximal independent sets in graphs with at
most $r$ cycles,'' with Goh Chee Ying, Koh, Khee Meng, and Bruce Sagan,
and ``Maximal and maximum independent sets in graphs with at most $r$
cycles, with Bruce Sagan, both accepted for J. Graph Theory), on packing
densities of permutations (``Pattern frequency sequences an internal 
zeros,''
with Mikl\'os B\'ona and Bruce Sagan and ``Bounding quantities related
to the packing density of $1(\ell+1)\ell\cdots 2$ with Martin 
Hildebrand and
Bruce Sagan, both published in Adv. Appl. Math.), on the partial 
well-order
theory of the permutation containment order (``Profile classes and 
partial
well-order for permutations,'' with Maximillian Murphy, published in
the Elec. J. Comb.), and on M\"obius
functions (``The M\"obius function of the composition poset,'' with 
Bruce Sagan
and still under consideration).

But Vince's work that is closest to my heart is his work on automated
combinatorics.  Vince began with a problem that no one thought could 
have
a nice answer.  He took a human approach to the counting of restricted
permutations (known as generating trees) and asked when they need only
finitely many labels.  Amazingly, he was able to prove necessary and
sufficient conditions for when this happens, and, more amazingly,
he developed a Maple package, {\bf FINLABEL}, which can find -- in a
completely automatic and yet rigorous way -- generating functions for 
such
classes.  His {\bf FINLABEL} package,
which accompanies his beautiful paper ``Finitely labeled generating 
trees for
restricted permutations,'' (J. Symb. Comp., accepted),
can reprove hundreds of results
(and several entire papers) that had previously been obtained my
hand.  This work is truly a fundamental breakthrough in the field
of restricted permutations, and has already inspired follow-up work
by others (see ``The insertion encoding of permutations,''
by Albert, Linton, and Ru\v{s}kuc, Elec. J. Comb., 12(1), R47).

Vince then took up an approach of my own.  I had developed the notion
of `enumeration schemes' to count more families of restricted
permutations, but was disappointed that
my technique had a success rate of only about 10\%. By
making highly ingenious {\it conceptual} improvements,
Vince extended my notion of {\it enumeration scheme} to include
many more cases.  His Maple package {\bf WILFPLUS} indeed is
a major enhancement of my previous package {\bf WILF}, and
his paper ``Enumeration schemes for restricted permutations''
(under consideration) is another major {\it theoretical}
advance in the field.

Of course, these techniques for automated combinatorics
apply to more than simply restricted permutations.
Recently Vince, Shalosh B. Ekhad, and I used
similar ideas
to settle an irresistible
conjecture of Nick Loehr and Greg Warrington that a certain
easy-to-describe set of lattice paths are enumerated by
$10^n$ in our paper
``A proof of the Loehr-Warrington amazing TEN to the power n 
conjecture''
(under consideration).  Our resulting Maple package, {\bf TEN},
was able to prove their conjecture in 30 seconds.
Since then, Loehr, Warrington and Bruce Sagan found a computer-free
proof for a generalization, while Jonas Sj\"ostrand gave a
beautiful bijective proof for an even broader conjecture.  But
the main point of our paper was to develop and illustrate a
methodology of completely computer-generated research,
which should extend to many other cases where the answers
are not so beautiful, and hence where it is very unlikely that there 
exist
nice human proofs.

On top of his fantastic research, Vince is also a very gifted teacher
and speaker.  Quite a few times he
has presented
excellent seminar talks.

Vince is a warm and enthusiastic person, as well as patient and
compassionate, all qualities that would make him very compatible with
Richard Stanley, the sponsoring scientist.

To sum up, Vince is an extremely promising young mathematician and a
gifted teacher. I recommend him most highly.

If you have any further questions, I will be glad to answer them.
My phone numbers are (609)921-7873 (home), and (732)445-1326 (office).

   \bigskip}
   \noindent
   \hskip 0.5\hsize\vbox to1.2truein{
    \hbox{Sincerely,}\vfil\hbox{Doron Zeilberger, Ph.D.}\hbox
{Board of Governors Prof. of Math}}
   %
   % Optional info you may want to include:
   %
   \rm\medskip
% \line{encl.\hfil}
% \line{ \hfil}
   %\line{ \hfil}
   \vfill
   \end





