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\frenchspacing

\auteurcourant={DOMINIQUE FOATA}
\titrecourant={THE LOTHARINGIAN IMPACT}


\vglue 3.5cm
\centerline{\bf The Lotharingian impact on Combinatorial
Mathematics:}
\centerline{\bf myth or reality?}
\bigskip
\centerline{Dominique Foata \footnote{$^1$}{The papers
which are electronically published on the S\'eminaire Lotharingien
site: {http://www.mat.univie.ac.at/$\sim$slc/} are referred to as
[SLC Bxx] and {\it not} further mentioned in the list of references at the end
of the paper.}}
\centerline{(Strasbourg)}

\vskip1cm
It is very fortunate that two, and sometimes three, sessions of the
S\'eminaire Lotharingien have been held every year since its
beginning in 1980. Otherwise, several of us, at least the confirmed
optimists, would have witnessed the jubilee in Paradise!
Anyway, this is a great privilege to have all of you here, at the
Domaine Saint-Jacques, a place regarded as a {\it valeur s\^ure}
by the younger generation.

In fact, the Seminar had long been in limbo, all through the
seventies, when regular meetings on Combinatorics were organized at
the Mathe\-ma\-ti\-sches For\-schungs\-ins\-ti\-tut,
Oberwolfach under the direction of Konrad Jacobs, a
distinguished probabilist, but also a great promoter of the
combinatorial methods in Mathematics. Suddenly, the Oberwolfach
Institute had become too small to shelter all the European
mathematicians interested in Combinatorics. It became urgent to
create a new structure, where everybody could come, bring his
contribution and be exposed to the current ideas of that field.

As had been told on several occasions, the Lotharingian Seminar had
a true spontaneous birth at Strasbourg in 1980, at a time when
Adalbert (a true Lotharingian Christian name!) Kerber kept shuttling
between Aachen (the seat of Charlemagne's kingdom) and Bayreuth.
Volker Strehl, already based in Erlangen, responded to the idea with
a great enthusiasm and accepted to be the third pillar of the whole
edifice. As a matter of fact, he immediately made the suitable
arrangements for the second session that took place in Burg
Feuerstein, near Bamberg, the same year, in the Fall 1980.

Since then, the three mother Universities, Bayreuth, Erlangen and
Strasbourg have scheduled two or three meetings per year,
occasionally assisted by other Institutions, such as Wien, Bologna,
Cagliari, Leoben, Potenza, Lyon. In 2000 a new trio took charge:
Wien-Erlangen-Marne-la-Vall\'ee, that soon became
Lyon-Erlangen-Marne-la-Vall\'ee, as Christian Krattenthaler has
moved from Austria to France. As everybody knows, together with
Volker Strehl and Jean-Yves Thibon, Christian rules over the
S\'eminaire with wisdom, science and efficiency.

Here is the full list of the sessions with their locations and their
organizing instances:
\medskip
{\parindent=30pt

\item{1980:}  01 Strasbourg (Strasbourg),  02 Burg Feuerstein
(Erlangen),

\item{1981:}  03 Le Kleebach (Strasbourg),  04
Bayreuth (Bayreuth),\hfil\break 05 Sainte-Croix-aux-Mines
(Strasbourg),

\item{1982:}  06 Burg Feuerstein (Erlangen),

\item{1983:}  07 Donndorf (Bayreuth),  08 Sainte-Croix-aux
Mines (Strasbourg), 09 Hollabrunn (Wien),

\item{1984:}  10  Burg Feuerstein (Erlangen),\hfil\break 11
Wasserschloss Mitwitz (Bayreuth),

\item{1985:}  12 Le Kleebach (Strasbourg),  13 Terme di Castel
San Pietro\hfil\break (Bologna),

\item{1986:}  14 Burg Feuerstein (Erlangen),  15 Schloss Schney
(Bayreuth),

\item{1987:}  16 Liebfrauenberg (Strasbourg),  17 Eremo San
Paolo  (Strasbourg),  18 Obsteig (Wien),

\item{1988:}  19 Schloss Schwanberg (Erlangen),  20 Alghero
(Cagliari),

\item{1989:}  21 Schloss Thurnau (Bayreuth),  22 Hesselberg
(Erlangen),

\item{1990:}  23 Rouge-Gazon (Strasbourg),  24 Liebfrauenberg
(Strasbourg),\hfil\break  25 Salzburg (Leoben),

\item{1991:}  26 Schloss Thurnau (Bayreuth),  27 Hesselberg
(Erlangen),

\item{1992:}   28 Domaine Saint-Jacques
(Strasbourg),\hfil\break    29 Schloss Thurnau (Bayreuth),

\item{1993:}  30 Hesselberg (Erlangen),  31 Domaine Saint-Jacques
(Strasbourg),

\item{1994:}  32 Domaine Saint-Jacques (Strasbourg),  33 Freiberg
(Bayreuth), 

\item{1995:}  34 Domaine Saint-Jacques
(Strasbourg), 35 Hesselberg (Erlangen),  

\item{1996:}  36 Schloss Thurnau (Bayreuth),  37
Strasbourg (Strasbourg),\hfil\break 38 Bellagio (Strasbourg),

\item{1997:}  37 Schloss Thurnau (Bayreuth),  40 Bellagio
(Strasbourg), 

\item{1998:}  41 Domaine Saint-Jacques
(Marne-la-Vall\'ee),\hfil\break   42 Maratea (Potenza),

\item{1999:}  43 Kloster Schoental (Erlangen).


\item{2000:} 44 Domaine Saint-Jacques (Strasbourg),\hfil\break 
 45 Bellagio (Marne-la-Vall\'ee),

\item{2001:} 46 Lyon (Lyon), 47 Bertinoro (Bologna),

\item{2002:}
48 Domaine Saint-Jacques (Marne-la-Vall\'ee),\hfil\break  49
Haus Sch\"onenberg, Ellwangen  (Erlangen),

\item{2003:}
50 Domaine Saint-Jacques (Marne-la-Vall\'ee
).
 
}


\medskip\noindent
The next meeting is already scheduled to take place in
Bertinoro.

\goodbreak
\medskip
Back in 1980 our intention
was to have a joint seminar for the three mother Universities, where
each group could discuss its recent works. Soon however, it appeared
useful to ask one of us, or a distinguished visitor, to prepare and
read a survey paper in Combinatorics, or more often in a related
area. 

We then had so-called main speakers. They have certainly enlarged
our vision of Combinatorics and enriched the connections between
several branches of Mathematics. Most of those survey papers have
been published in our proceedings, in paper form in the beginning,
then in electronic form after 1994. I will not speak about the
change of policy of our S\'eminaire when we decided to move from
traditional to electronic publishing, but will focus my attention to
the main mathematical themes that were discussed along those
years, after mentioning a few basic facts and figures.

\bigskip
\centerline {\bf 1. Some statistics}

\medskip
{\bf 50 sessions}, so far, as we celebrate the jubilee.

{\bf 27 locations} used for the S\'eminaire, 
the Domaine Saint-Jacques (8 times), Thurnau (5 times), Burg
Feuerstein and Hesselberg (each 4 times), Bellagio (3 times),
Strasbourg, Kleebach, Sainte-Croix-aux-Mines, Liebfrauenberg
(each twice), the other places once.

{\bf 23 years of existence}, so far, from 1980 to 2003.

{\bf 29 printed seminar proceedings}: a joint issue 1-3, all issues
from no.~5 to no.~31, plus no.~34, published in the series {\it
Publications de l'Institut de Recherche Math\'ematique Avanc\'ee,
Strasbourg}.

{\bf 323 papers, electronically deposited} on the S\'eminaire
Lotharin\-gien site:
{\tt http://www.mat.univie.ac.at/$\sim$slc/}.

{\bf 332 registered Lotharingians} listed on the site, mostly
mathematicians who have attended the S\'eminaire at least once.
Almost one paper per Lotharingian!

\bigskip

\centerline{\bf 2. The impact?}\medskip

If we examine the classification of Mathematical Reviews to-day, we
notice that 05 Combinatorics is subdivided into five subclasses:


\medskip
    05A (1973 -Current ) Enumerative combinatorics 

    05B (1973 -Current ) Designs and configurations 
    
    05C (1973 -Current ) Graph theory 
    
05D (1991 -Current ) Extremal combinatorics 
 
05E (1991 -Current ) Algebraic combinatorics 
         
\medskip
We have to admit that few papers on Designs, Graphs and Extremal
Problems have been read in the sessions of the S\'eminaire, with the exception
of Matroid Theory (essentially by the Bielefeld school), waiting
for the enumerative aspects of that theory by Bodo Lass, the study
of polyominoes by the Bordeaux school, Ramsey Theory by the late
Walter Deuber, a fascinating introduction to the four-colour
problem [Co87] with many illuminating figures by Daniel I. A.
Cohen, a paper that would deserve an electronic version.

\medskip
As for 05A Enumerative and 05E Algebraic Combinatorics, they have been
most present in our S\'eminaire. Mathematical Reviews provides the further
descriptions: 

\medskip
     \halign{#\hfil\cr    
             05A05 (1973-now) Combinatorial choice problems \cr
     \hskip 3cm(subsets, representatives,
permutations)\cr
             05A10 (1973-now) Factorials, binomial coefficients, \cr
\hskip 3cm combinatorial functions \cr
             05A15 (1973-now) Exact enumeration problems, generating functions \cr
             05A16 (1991-now) Asymptotic enumeration\cr
             05A17 (1973-now) Partitions of integers \cr
             05A18 (1991-now) Partitions of sets\cr
             05A19 (1973-now) Combinatorial identities\cr
             05A20 (1973-now) Combinatorial inequalities\cr
             05A30 (1985-now) $q$-calculus and related topics \cr
             05A40 (1985-now) Umbral calculus\cr
     }

\goodbreak
\medskip
     \halign{#\hfil\cr 
             05E05 (1991-now) Symmetric functions\cr
             05E10 (1991-now) Tableaux, representations of the symmetric group\cr 
             05E15 (1991-now) Combinatorial problems concerning \cr
    \hskip 3cm the classical groups \cr
             05E20 (1991-now) Group actions on designs, geometries and codes\cr
             05E25 (1991-now) Group actions on posets and homology \cr
\hskip 3cm groups
of posets\cr 
             05E30 (1991-now) Association schemes, strongly regular graphs\cr
             05E35 (1991-now) Orthogonal polynomials\cr 
            }

\medskip
Some items of 05A Enumerative Combinatorics seem to be old-fash\-ioned and
there is some disparity of topics within 05E Algebraic Combinatorics. Anyway,
let us take them as such and examine how they have been illustrated along our
sessions. Of course, my description will be awfully sketchy and partial, but
there was no way ``to cram within this short address the vasty fields" of
Lotharingian Combinatorics, to paraphrase King Henry the Fifth's Chorus. 

\bigskip
As I cannot make an exhaustive excursion to all the topics, I will shortly
mention some topics that were regularly discussed. Again, I should like to
apologize myself for being so brief on so many fascinating and
fast-growing subjects. Let me mention:

\medskip
The {\it theory of permanents} and its
challenging conjectures, thanks to the expertise of Arnold Kr\"auter (see his
three papers [SLC B09b, B11b, B16b]), Norbert Seifter and Peter Gerl and
the more recent paper by Guoniu Han and Christian Krattenthaler on the
Scott-type permanents [SLC B43g]. 

The theory of
{\it Codes and Languages}, presented by our colleagues Berstel,
K\"onig and Perrin, in particular; automata studies by Werner Kuich
[Ku88], V\'eronique Bruy\`ere [SLC B35b], Jean-Pierre Allouche [SLC B30c] in
relation with arithmetical problems.

{\it P\'olya theory} and its impact on Chemical Engineering by Adalbert
Kerber.


{\it Maps and hypermaps} by Robert Cori, Didier Arqu\`es.

{\it Plane tiling} by
Andreas Dress and his school, especially {\it hexagon tiling} by Ilse Fischer
[SLC B45f] and the Viennese school.

{\it The theory of Witt vectors} by Henri Gaudier
[SLC B18c, B21f] and Andreas Dress [SLC B28c].

{\it Algorithmic Complexity} by Peter B\"urgisser
and Michael Clausen [SLC B36a, B36b].

{\it Umbral Calculus} by Marilena Barnabei and her
Bolo\-gnese co-workers; also by Luigi Cerlienco and his Sardinian
co-workers.

{\it Tree Enumeration} by
Peter Kirschenhofer [Ki90]. The paper exists in a pre-\TeX\ form. Peter 
should be convinced to offer us a definitive electronic
version.

\medskip  
Next, let me  present a few Lotharingian gems I should like to extract from
our Treasure.

\bigskip
 \centerline{\bf 3. Enumerative Combinatorics}

\medskip
In 1988 Volker Strehl [St88] brought our attention to the following 
identity due to P.~Brock [Br60]: for $A,B\ge 0$ let
$$\displaylines{
H(A,B)=\sum_{i=0}^A\sum_{j=0}^B
{i+j\choose j}{A-i+j\choose j}{B+i-j\choose B-j}{A-i+B-j\choose B-j}.\cr
\hbox{Then}\qquad
H(A,B)-H(A-1,B)-H(A,B-1)={A+B\choose A}^2.\hfill\cr}
$$
Such identities are no mystery to-day, as they can be proved by means of
computer algebra methods (Zeilberger, Paule, Krattenthaler), but what Volker
taught us in 1988 was an implementation of the identity into various
combinatorial contexts, in particular in his powerful {\it periodical,
local-injective endofunction} geometry (the PLI-endofunctions), in which the
identity reflected a banal counting. 

With his kind permission the graph of
such a PLI-endofunction is reproduced in Fig.~1. First, an ordered partition
$(S_1,S_2,\ldots, S_p)$ (here $p=3$) of a finite set~$S$ is given; then,
for each $i=1,2,\ldots,p$ the restriction of the endofunction to~$S_i$ is
an injection of $S_i$ into $S_i\cup S_{i+1}$ (by convention
$S_{p+1}:=S_1$). Thus, the name of {\it local-injective} and {\it
periodical} is very appropriate. Next, such an endofunction can be given a
{\it weight} that takes the cycles within each block~$S_i$ into account
and also the cardinalities of the~$S_i$'s. For instance, the
PLI-endofunction in Fig.~1 is given the weight
$$
(1+\alpha_2)^2(1+\alpha_3)x_1^8x_2^{11}x_3^{10},
$$
as there are two cycles in~$S_2$ and one cycle in $S_3$ and since $S_1$,
$S_2$ and $S_3$ are of cardinalities 8, 11, 10, respectively.

\midinsert
$$\epsfbox{pub93Fig.1}$$ % For PostScript
%$$\convertMPtoPDF{pub93Fig.1}{1}{1}$$ % For PDF
\centerline{Fig. 1}
\endinsert

The problem left to the reader is to use the above algebra of the
PLI-endofunctions with their weights and derive the Brock identity. The
paper by Volker Strehl (hopefully in electronic form some day) will certainly
rescue the unsuccessful reader in case of difficulty.

\medskip
Let me  digress on the paper by Dumont [SLC B05b] that
updates an old paper by Seidel [Se77] on the finite-difference
calculus of classical numbers, because simple facts in mathematics are
recurrent and deserve to be remembered. Let
$(a(0,n))$ $(n\ge 0)$ be a given sequence, called the {\it initial
sequence}, of elements from a ring
$\Omega$ and define the matrix $(a(i,j))$ $(i,j\ge 0)$ by:
$$
\vbox{\halign{\hfil$#={}$&$#$\hfil&\qquad$#$\hfil\cr
a(i,j)&a(i-1,j)+a(i-1,j+1)&(i\ge 1,\,j\ge 0).\cr
}}
$$
The {\it leftmost column} $(a(n,0))$ $(n\ge 0)$ of the above matrix
is called the {\it final sequence}. It is easy to derive the generating
function for the latter sequence from the generating function for the
initial one. There are several unexpected correspondences between
those two sequences. For instance, let $E_n$ $(n\ge 0)$ be the
numbers defined by the generating function
$$
\tan u+\sec u=\sum_{n\ge 0}{u^n\over n!}E_n
=1+{u\over 1!}1+{u^2\over 2!}1+{u^3\over 3!}2
+{u^4\over 3!}5+{u^5\over 5!}16+\cdots
$$
so that the coefficients $E_{2n-1}$ are the {\it secant numbers}
and $E_{2n}$ the {\it tangent numbers}.
If we take $a(0,0)=1$,
$a(0,2n)=0$ $(n\ge 0)$ and $a(0,2n-1)=(-1)^nE_{2n-1}$ (the
signed tangent number), then the final sequence reads:
$a(2n,0)=(-1)^nE_{2n}$ (the signed secant number) and
$a(2n-1,0)=0$. 

\bigskip
\centerline{\bf 4.  $q$-calculus and related topics}
\medskip
The algebra of
$q$-series goes back to Heine (1850) and has long be nurtured
by the specialists of Partition Theory, such as our friend George
Andrews. It entered the field of Enumerative Combinatorics in the
early seventies with the works of Carlitz, Gessel in his Ph.D. thesis
[Ge77], although many calculations were anticipated by MacMahon
himself [MacM15] in his monumental treatise. 

The first issue of the S\'eminaire ever published came with the fifth
Session that took place in the School of Music in Sainte-Croix-aux-Mines in
1981. It contains a superb contribution of J. Cigler [SLC B05a] to the study
of $q$-Calculus, which is modestly called ``Elementary $q$-Identities". His
operator approach is very convincing and deals with the algebra of
$q$-Hermite and $q$-Laguerre polynomials that enables him to derive a
simple proof of the Mehler formula for the $q$-Hermite polynomials. 

Let me mention that identity by recalling, as was done by Jacques
D\'esarm\'enien [SLC~B06b], that several versions of those polynomials exist
but can be transformed into one another by a suitable change of variables.
We had to wait for 2002 to have a true combinatorial proof of that famous
identity, a proof due to Hung Quang Ngo [SLC B48b], a student of Dennis
Stanton.

In the sequel we make use of the usual notations:
$$
\eqalign{(a;q)_n&=\cases{1,&if $n=0$;\cr
(1-a)(1-aq)\,\ldots\,(1-aq^{n-1}),&ifi $n\ge 1$;\cr}\cr
(a;q)_{\infty }&=\lim\nolimits_n(a;q)_n
=\prod_{n\geq 0}(1-aq^n).\cr}
$$
One of the versions of the $q$-Hermite polynomials $H_n(x|q)$ can be
defined by their generating function
$$
\sum_{n\ge 0} H_n(x|q) {u^n\over (q;q)_n}
=\prod_{k\ge 0}{1\over (1-2xuq^k+u^2q^{2k})}.
$$
Then Hung Quang Ngo was able to derive the identity
$$
\displaylines{\quad
\sum_{n\ge 0} H_n(x|q) H_n(y|q){u^n\over (q;q)_n}\hfill\cr
\noalign{\vskip-10pt}
\hfill{}=
\prod_{k\ge 0}
{(u^2;q)_\infty \over
(1-4uq^kxy+2u^2q^{2k}(-1+2x^2+2y^2)
-4u^3q^{3k}xy+u^4q^{4k})},\cr\quad}
$$
using a very clever combinatorial set-up of {\it bicolored
$(q,n)$-involutionary graphs}.

Let me go back again to that first issue and mention the paper by
Daniel Barsky [SLC B05b] on the $p$-adic analysis of the sequences
of classical numbers. It is a pity that the $p$-adic approach has
not been used more extensively in combinatorial studies, as it was
a powerful tool in providing a short proof [Ba81] of the Gandhi
conjecture [Ga70] for the Genocchi numbers.


The paper by Josef Hofbauer on Lagrange-Inversion has been electronically
reproduced [SLC B06a]. It contains the various $q$-versions of the Lagrange
inversion identity, mentions the multilinear extension (due to Good but
already known to Jacobi, as Hofbauer found out) and shows how the MacMahon
Master Theorem is a plain consequence of that multilinear extension. This
raises the question of imagining a true $q$-multilinear version of the
Lagrange identity (perhaps it is not even an identity), that would be a
handy tool in the algebra of $q$-series. Notice that a very convincing
combinatorial proof of the multivariable Lagrange inversion formula is due
to Gessel [Ge87] and  a $\beta$-extension was derived by Zeng [SLC B20b].

As Pierre Cartier puts it [SLC B23a] there are ``new adventures in the
country of $q$-calculus" in Statistical Mechanics and the Yang-Baxter
equation. 

\bigskip
\goodbreak
\centerline{\bf 5. Partitions of integers}
\nobreak
\medskip
Back in the seventies the classical theory of partitions had not yet exploded.
It was most challenging to wait for the latest contribution of George Andrews
to make sure that Partition Theory had not become a foreign subject to
most of us. Regretfully we only have two contributions of his in our
proceedings [SLC B25f, B42i], the second one jointly written with Peter
Paule on the rebirth of MacMahon's partition analysis. In Maratea George
Andrews and Peter Paule [SLC B42i] read their fourth paper on the subject
(``MacMahon's Partition Analysis IV: Hypergeometric Multisums"). 
Other papers on partitions appear in vol.~42, the Andrews Festschrift, that
contains a formidable paper by Bruce Berndt and Ken Ono on the
tau-function and the work of Ramanujan. Thanks to Peter Paule [SLC B18f]
the Bailey chain tool for proving $q$-identities has become a classic.


\bigskip
\centerline{\bf 6. Symmetric Functions}

\medskip
 This is a topic that has been
extensively studied, especially under the leadership of Alain
Lascoux, a strong symmetric function believer. When the
proceedings of the eighth session were published (Publ. {\sevenrm
IRMA}, 1984 229/S-08), he had already written, jointly with the late Marco
Sch\"utzenberger, his first memoir on the subject [La84]. Only the
first chapter of that memoir was printed in the proceedings. The good news is
that an updated version of the memoir has just appeared in print [La03].

Let me mention the paper by Ira Gessel [SLC B17a] that extends the notion of
D-finiteness to the class of symmetric functions and makes a clever use of that
notion to derive the P-recursiveness of several combinatorial
sequences.\footnote{$^2$}{ I owe this informative comment to Christian
Krattenthaler, who added that Gessel's approach has been taken up again by Chyzak,
Mishna and Salvy in a recent work.}

In 1988 we were privileged to be invited by the Sardinian School to have the
S\'eminaire at Alghero. We could have Ian G. Macdonald as a main speaker,
who, for the first time, read his seminal paper on ``A new class of symmetric
functions" [SLC B20a]. For each positive integer~$r$ let $p_r$ denote the
power-sum $\sum x_i^r$ and for each partition
$\lambda=(\lambda_1,\lambda_2,\ldots\thinspace)$ let
$p_\lambda:=p_{\lambda_1}p_{\lambda_2}\ldots$ and denote by
${\ell}(\lambda)$ the number of parts of~$\lambda$; furthermore, if the
multiplicative form of~$\lambda$ reads $\lambda=1^{m_1}2^{m_2}\ldots$,
let $z_\lambda:=\prod_{r\ge 1} (r^{m_r}\cdot m_r!)$. Finally, define
$$
z_\lambda(q,t):=z_\lambda\prod_{i=1}^{\ell(\lambda)}
{1-q^{\lambda_i}\over 1-t^{\lambda_i}}.
$$
In the
algebra of symmetric functions Macdonald introduced the scalar product
$\langle\ ,\rangle_{(q,t)}$ defined by
$$
\langle p_\lambda,p_\mu\rangle_{(q,t)}=\delta_{\lambda\mu}
z_\lambda(q,t)
$$
and proved that there is a unique class of symmetric functions
$P_\lambda=P_\lambda(q,t)$ with coefficients in the field of rational
functions in~$q$ and~$t$, which are orthogonal with respect to the above
scalar product. Moreover, those polynomials specialize to all known classical
bases of the symmetric functions: the monomial, homogeneous, elementary,
Schur, Hall-Littlewood, Jack functions. 

Several conjectures concerning those
polynomials were stated at that time. The paper has been the source of 
numerous significant works. In particular, Jennifer Morse [SLC B41a] 
was able to give an explicit expansion of $P_\lambda$ in the case
of two variables. It is worth mentioning that the Lotharingian memoir itself
was later reproduced, with minor changes, as the sixth chapter in the
second edition of Macdonald's book [Mac95]. 

The first meeting at Domaine Saint-Jacques took place in Spring 1992. It was
the twenty-eighth session. Macdonald [SLC B28a] brought a further
contribution to the subject by unifying various recent extensions and
analogues of Schur functions, especially the ribbon Schur functions
introduced by Lascoux and Pragacz [La88]. 

A fruitful application of the Schur function algebra was made by Jacques
D\'esarm\'enien [SLC B15a] who reproved the classical Gordon and
Macdonald identities and also derived two new identities, regarded as
$q$-analogs of two formulas obtained by Desainte-Catherine and Viennot on
partitions. Those two $q$-identities read:
$$ \displaylines{
\sum _{\lambda\subset (2m)^n} 
S_\lambda(q^n,q^{n-1},\dots,q)=\prod
_{1\le i\le j\le n} 
{{q^{2m+i+j}-1}\over  {q^{i+j}-1}};\cr
\sum _{\lambda\subset (2m)^n}\kern-8pt
S_\lambda(q^{2n-1},q^{2n-3},\dots,q)\kern-4pt
=\prod _{1\le i\le n} { {q^{2m+2i}-1}\over {q^{2i}-1}}
\kern-4pt\prod
_{1\le i<j\le n}{ {q^{2(2m+i+j)}-1} \over {q^{2(i+j)}-1}};
\cr}$$
where the sum is over all partitions
 $\lambda$ having only even parts.

\bigskip
\centerline{\bf 7. Orthogonal polynomials}

\medskip
Let $(p_n(x))$ be a sequence of orthogonal polynomials with respect to a
functional~$\cal L$. For several classical polynomials (Hermite, Laguerre,
Krawtchouk, Meixner, \dots) the evaluation of the functional
${\cal L}(\prod_{i=1}^m p_{n_i}(x))$ was to be made and the major
problem was to characterize the positivity domain of that functional.

The combinatorial approach consists of
reinterpreting the functional as a generating function for 
combinatorial structures, such as muticolored derangements,
by some univariable or multivariable statistics. If a {\it closed} formula
can be derived for the functional, the positivity property becomes
a simple by-product or a bonus! That approach has been successfully used
by Jiang Zeng in several contributions; see [Zen88] in particular.

\bigskip
\centerline{\bf 8. Tableaux, representations of the symmetric group}

\medskip
Let me mention the Robinson-Schensted analog for oscillating tableaux
derived by Marie-Pierre Delest and her followers [SLC B20b], the geometric
version of the Kerov-Kirillov-Reshetikhin construction by Guoniu Han
[SLC 31a] and a study of tableaux of dominoes by Christophe Carr\'e and
Bernard Leclerc [SLC B31c]. The celebrated {\it jeu de taquin} was
restructured by Marc van Leeuwen [SLC B41b] to walk over the
Crystal paths of Littelmann.

In Representation Theory proper Adalbert Kerber and his school have
constantly be present, and also the Marne-la-Vall\'ee phalanst\`ere [SLC B32c]
and Christine Bessenrodt [B33a].

\bigskip
\centerline{\bf 9. Computer-assisted proofs of identities}

\medskip
In 1993 we invited Doron Zeilberger to give three talks on his fantastic
method of proving hypergeometric identities with the help of a computer. We
were sure that his lectures would be fascinating and probably most
entertaining (they were!), but were afraid of learning
more on the scientific achievements of his chairman compared with the
talent and production of his personal computer than about hypergeometric
identity proving. To our great surprise his lectures were masterly done and
the text he gave us [SLC B24b] remains an excellent introduction to the
subject, even to-day. An updated version of the paper was further published
[Ze95].

In the same vein Volker Strehl [B29b] studied the conjugacy of sequences
satisfying linear recurrences by having a closer look at the identity
$$
\sum_{k=0}^n{n\choose k}^2{n+k\choose k}^2
=\sum_{k=0}^n {n\choose k}{n+k\choose k}\sum_{j=0}^k {k\choose j}^3,
$$
an identity that involves the famous Ap\'ery numbers
$\sum_k{n\choose k}^2{n+k\choose k}^2$. Volker showed us how the interplay
between computer algebra softwares and some human cleverness (!) can lead to
a new approach of diophantine approximation.

Pierre Cartier's Mathemagics [SLC B44d] convinced us that he remained a
strong supporter of a sound use of experimental mathematics. We
had expected a memoir on the polylogarithms and their connections
with the shuffle algebras, as was developed by the Lille school, but
it would have been of double use with the two papers on the
subject read by Minh [SLC B43e, B44i].

We are finally proud of saying that the first announcements of the
implementations of Christian Krattenthaler's HYP and HYPQ packages [SLC
B30a] were first made in a session of our S\'eminaire (the thirtieth).

\bigskip
\goodbreak
\centerline{\bf 10. Determinant Calculus}
\nobreak
\medskip
Let me quote George Andrews in his review [MR2002i:05013] of
Christian Krattenthaler's 67-page paper ``Advanced Determinant
Calculus" published in [SLC B 42q]: ``This is a valuable, useful and
unique survey carefully prepared by the only person who could
successfully pull it all together." Yes indeed, many evaluations in
Combinatorics first to lead to a determinant that remains to be
calculated, but how? We feel privileged to have that unique {\it
oiseau rare} among us. See also his paper on $q$-analogues of
Determinant Identities arising in Plane Partitions [SLC 36e] and
[SLC 47g].

\bigskip
\centerline{\bf 11. Other excursions}

\medskip
The S\'eminaire had welcomed the incursion into Probability Theory
with the contribution of Roland Speicher [SLC B39c] (``Free probabililty
theory and non-crossing partitions"). I will not explain the first
part of the title, but rather recall what the second part refers to.
Here, partition means set-partition, and more precisely a set of
disjoint non-empty blocks whose union is the interval
$[n]=\{1,2,\ldots, n\}$. If there exist four integers $1\le
i<k<j<l\le n$ such that 
$i$ and $j$ belong to the same block and $k$ and $l$ belong to
another block of a partition~$\pi$, we say that $\pi$ is a {\it
crossing} partition. Otherwise, $\pi$ is said to be {\it
non-crossing}. Those non-crossing partitions of $[n]$ form a
lattice, that is quite similar to the lattice of {\it all} partitions
of $[n]$. It is remarkable that the notion first introduced and
studied by Kreweras [Kr72] was exactly the object needed by the probabilists
to express the relations between the moments of a random variable
and the so-called free cumulants.

\bigskip
\centerline{\bf 12. Statistical Study of Coxeter Groups}

\medskip
The purpose is to see how various statistics defined on the symmetric
group, such as the descent number, the length, the major index\dots\ can be
naturally (?) extended to certain Coxeter groups and how reasonable (?)
analytic expressions can be derived for the generating series for those
statistics. Back in the seventies the difficulty was to imagine the
appropriate normalization for the series involved. For instance, the
coefficients~$C_n$ in the following identity
$$
\sum_n 
{u^n\over (s;p)_{n+1}\,(t,q)_{n+1}} C_n
=\sum_{k,l} s^kt^l
{(-zu;p,q)_{k+1,l+1}
\over (u;p,q)_{k+1,l+1}}
$$
can be interpreted as a generating {\it polynomial} for permutations by
some multivariable statistic. Observe the normalization of the series.
On the right-hand side $(u;p,q)_{k+1,l+1}$ stands for the product 
$\prod(1-up^iq^j)$ $(0\leq i\leq k; 0\leq j\leq l)$.
Such a normalization for this kind of problem goes back to Gessel in
his Ph.D. thesis and was essential for expressing further permutation
statistic identities.

Recently, the Israeli school with Ron Adin, Francesco Brenti,
Yuval Roichman [Ad03] used a further normalization
$$[t;{\bf q}]_n:=\cases{1-t,&if $n=0$;\cr
(1-t)(1-tq_1)(1-tq_1q_2)\cdots (1-tq_1\cdots q_n),&if $n\ge
1$,\cr}
$$
so that we can derive:
$$
\sum _n  
{u^n\over [s;{\bf p}]_{n+1}\, [t;{\bf q}]_{n+1}}
C_n(z,s,t,{\bf p},{\bf q})
=
\phi_p\phi_q\sum_{k,l}s^kt^l\kern-8pt
\prod _{\scriptstyle 1\le i\le k+1\atop \scriptstyle
1\le j\le l+1}{(1+zux_iy_j)\over
(1-ux_iy_j)}.
$$
In that formula $\phi_p$, $\phi_q$ are linear homomorphisms, but not ring
homomorphisms. Further work is to be done to derive an extension of the next
to the last identity with the elimination of $\phi_p$ and $\phi_q$.

Several transformations on the symmetric group have been constructed and
were used to prove that certain classical statistics, such as the major index
and the inversion number, were equidistributed. Analogs of those
transformations should be constructed, for other Coxeter groups, at least
for the signed permutations, that would help prove further
equidistribution properties.

\bigskip
\centerline{\bf 13. Conclusion}

\medskip
Nobody would object that Lotharingia as a whole---that is,
the very many participants in our S\'eminaire---has had a real
impact on Combinatorics. Our S\'eminaire was created at the time 
of the explosion of that discipline, thus opening it up to a challenging
exchange from the very beginning. Moreover, the door has been kept open to
other fields that welcome combinatorial methods. A new generation of
mathematicians is now in command of both the organization of the seminars
themselves and of the electronic journal. They keep a watchful eye on the
frontiers: Classical and Commutative Algebra, Computer Algebra, Special
Functions, Theoretical Computer Science, Algorithms, Probability, \dots\  There
is no doubt that our S\'eminaire is more vivid than ever.


\goodbreak
\vfill\eject
\vglue 2.2cm

\centerline{REFERENCES}
\bigskip

\divers Ad03|Ron M. Adin, Francesco Brenti, Yuval Roichman|Descent
Representations and Multivariate Statistics, preprint 47 pages|


\divers Ba81|Daniel Barsky|Congruences pour les nombres de
Genocchi de deux\-i\`eme esp\`ece, Groupe d'\'etude d'analyse
ultram\'etrique (Y.~Amice, G.~Christol, P. Robba), 8\ieme~ann\'ee,
{\oldstyle1980}/{\oldstyle 81}, n\omini~34, 13~p. Secr\'etariat de
math., Univ. Paris~7|

\article Br60|P. Brock|On a binomial identity arising from a sorting problem,
problem 60-2|SIAM Review|2|1960|40|


\article Co87|Daniel I. A. Cohen and V.
S. Miller|On the four color problem|Publ. {\eightsl IRMA} Strasbourg|
341/S-16|1987|5-62|

\article Ga70|Gandhi (J.M.)|A conjectured representation of
Genocchi numbers|Amer. Math. Monthly|77|1970|505--506|

\divers Ge77|Ira Gessel|Generating functions
and enumeration of sequences, Ph. D. thesis,
Dept. Math., M.I.T., Cambridge, Mass.,
111 p., {\oldstyle 1977}|

\article Ge87|Ira Gessel|A combinatorial proof of the multivariable
Lagrange inversion formula|J. Combin. Theory Ser. A|45|1987|178-195|

\article Ki90|Peter Kirschenhofer|Enumeration of trees|Publ. {\eightsl IRMA}
Strasbourg|462/S-25|1990|19-61|

\article Kr72|Germain Kreweras|Sur les partitions non-crois\'ees d'un
cycle|Discrete Math.|1|1972|333-350|

\article Ku88|Werner Kuich|Semirings, automata and combinatorial
applications|Publ. {\eightsl IRMA}
Strasbourg|358/S-18|1988|5-51|

\livre La03|Alain Lascoux|Symmetric Functions and Combinatorial Operators on
Polynomials|Providence, R.I., Amer. Math. Soc., CBMS, vol.~{\bf 99} {\oldstyle
2003}|

\article La88|Alain Lascoux and Piotr Pragacz|Ribbon Schur functions|Europ.
J. Combinatorics|9|1988|561-574|

\livre La84|Alain Lascoux and M.-P. Sch\"utzenberger|Formulaire
raisonn\'e des fonctions sym\'etriques|L.I.T.P., U.E.R. Math., Univ.
Paris VII, 138~p., {\oldstyle 1984}|

\livre Mac95|Ian G. Macdonald|Symmetric Functions and Hall
polynomials. {\rm Second Edition}|Oxford, Clarendon Press, {\oldstyle 1995}|

\livre MacM15|Percy Alexander MacMahon|Combinatory Analysis, {\rm vol.~1
and~2}|Cambridge, Cambridge Univ. Press, {\oldstyle 1915}, 
(Reprinted by Chelsea, New York, {\oldstyle 1955})|

\divers Se77|L. Seidel|\"Uber eine einfache Entstehungsweise der
Bernoullischen Zahlen und einiger verwandten Reihen, {\sl
Sitzungsberichte der mathe\-matisch-physikalischen Classe der K\"oniglich
Bayerischen Akademie der  Wissenschaften. M\"unchen}, {\oldstyle 1877},
p.~157-187|

\article St88|Volker Strehl|Endliche Funktionen und spezielle Funktionen:
eine kombinatorische Theorie zu einer Identit\"at von P. Brock und deren
Erweiterungen|Publ. {\eightsl IRMA}
Strasbourg|348/S-17|1988|23-67|

\article Ze95|Doron Zeilberger|Three recitations on Holonomic Systems and
Hypergeometric Series|J. Symbolic Comp.|20|1995|699--724|


\article Zen88|Jiang Zeng|Lin\'earisation de produits de polyn\^omes de
Meix\-ner, Krawtchouk et Charlier|Publ. {\eightsl IRMA}
Strasbourg|348/S-17|1988|69-95|





\bigskip\bigskip
\rightline{\vtop{\halign{#\hfil\cr
Dominique Foata \cr
Institut Lothaire\cr
1, rue Murner\cr
F-67000 Strasbourg, France\cr
\noalign{\smallskip}
{\tt foata@math.u-strasbg.fr}\cr}}\hskip 1cm}
















\bye



------------=_3F954266.4C191AB6--
