%lara.tex:

%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
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\font\eightrm=cmr8 \font\sixrm=cmr6
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\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\1{{\overline{1}}}
\def\2{{\overline{2}}}
\parindent=0pt
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\def\frac#1#2{{#1 \over #2}}
%\headline={\rm  \ifodd\pageno  \RightHead  \else  \LeftHead  \fi}
%\def\RightHead{\centerline{
%Title
%}}
%\def\LeftHead{ \centerline{Doron Zeilberger}}
%end macros
\bf
\centerline
{
Request for an NSF supplement for the AY 2006-07 for Graduate Student
Support
}
\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  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg} .
}
}

I would like to request a supplement to support a graduate student, Lara
Pudwell (see CV attached), to work on the part of the current proposal
dealing with restricted permutations and enumeration schemes. Lara, who is
currently a third-year graduate student, is perfectly suited for pursuing
this research and has already done significant work. See, for example, her
completely computer-generated web book ``Systematic Studies in Pattern
Avoidance'' 

{\eighttt http://www.math.rutgers.edu/~lpudwell/webbook/bookmain.html .}

Besides the pioneering work by H. Wilf, A. Burstein and T. Mansour, all
dedicated to special cases, not much is known. In particular, all the work
done so far was done in the traditional, computer-free, mode. 

Moreover, some of this work is laden with minor but irritating
errors, and Lara already systematically corrected all of them.
To cite just one example, Lara corrected formula (6) of
Alex Burstein and Toufik Mansour's paper ``Words restricted
by patterns with at most 2 distinct letters'' (Elec. J. Comb. 9(2),
\#R3 (2002)) from the erroneous
$$
F_{112}(x,y)= {{1} \over {1-y}} \cdot 
\left ( {{1-y} \over {1-y-xy}} \right )^{1/y}
$$
to the corrected
$$
F_{112}(x,y)=
{{y} \over {(1-y)^2}} \cdot
\left (
(1-y-xy)
  \left ( 
{{1-y} \over {1-y-xy}}
   \right )^{(1+y)/y}
+y-1
\right )
$$

I believe
that using the method of rigorous experimental mathematics will further
this field significantly.  In addition, having the release-time from
teaching that this supplement will grant will be very instrumental for the
sponsored project, and will also be conducive to Lara's professional
growth.
\end









