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\centerline{\bf DESCRIPTION OF THE STRASBOURG GROUP}

\medskip

\bigskip

{\bf A. The group itself.}

\medskip
In the field of Algebraic Combinatorics there are six people in
Strasbourg enrolled in the EEC Human Capital and Mobility
Programme, namely

\smallskip
{\parindent=10pt
Dominique Dumont, Dr Sc., ma\^\i tre de conf\'erences;

Dominique Foata, Dr Sc., professor;

Guo-Niu Han, Dr, charg\'e de recherches {\sevenrm C.N.R.S.};

Arthur Randrianarivony, Dr, research fellow;

Yaacob Akiba Slama, doctoral student;

Jean Zeng, Dr, ma\^\i tre de conf\'erences.

}
\smallskip
To those six researchers we should add three other colleagues
who attend the  weekly Strasbourg Combinatorics seminar and 
also the biennial {\sl S\'eminaire Lotharingien de Combinatoire}
(Bayreuth, Erlangen, Strasbourg): 

\smallskip
{\parindent=10pt
Abdallah Al Amrani, Dr Sc.,
ma\^\i tre de conf\'erences, 

Jean-Pierre Jouanolou, Dr Sc.,
professor

Raphaelle Supper, Dr, ma\^\i tre de
conf\'e\-rences,

}
\smallskip
whose main research interests are centered
around Algebraic Geometry for the first two and Classical
Analysis for the last one.

\medskip
During the two-year period 1994-95 the group has benefited
from the EEC Human Capital and Mobility Programme with the
invitations of Dr Einar Steingr\`\i msson (Goteborg), in
Strasbourg for two months, and of Dr Christian Krattenthaler
(Wien) who only stayed two weeks.

\medskip
The group has also taken advantage of the invitations of Prof.
Doron Zeilberger (Philadelphia), Mike Hirschhorn (Sydney) and
Robert J. Clarke (Adelaide). The three of them have been
appointed visiting professors in Strasbourg, respectively for
one, two and four months. As will be noted below there had been
a strong research interaction between them and the Strasbourg
group.

\bigskip


{\bf B. Research orientations.}

\medskip
The research of the group has four main directions :
Combinatorics of classical numbers,
Combinatorial Study of Special Functions,
$q$-Eulerian Calculus, Analytical Tool Study.
However, those topics are strongly interconnected.

\medskip
{\it  $1$. Combinatorics of classical numbers}.\quad This study
goes back ot the early seventies when several schools
discovered the underlying combinatorial properties of classical
numbers : Euler, tangent, Genocchi numbers\dots\ Several
questions had remained unsolved, for instance, whether a joint
combinatorial model for both Genocchi and median Genocchi
numbers could be found. The problem has been completely solved
by Dumont and Randrianarivony [8]. The solution has suggested
several extensions, in particular the study of new classes of
polynomials that appear as generating polynomials for
statistics on finite structures counted by those numbers, or
simply as polynomials that specialize to those numbers and to
other related classical numbers (see, e.g., [5, 9, 10, 20, 23, 24, 25,
27]).

\smallskip
In several instances, exponential generating functions for
those numbers or their polynomial extensions cannot be
evaluated, but {\it continued fraction}
expansions for their ordinary generating functions can be
evaluated. The combinatorial theory of formal continued fractions
is well-known to-day, but in each example, an adequate
calculation is to be found. The most interesting result is
probably due to Randrianarivony [21] who could introduce true
$q$-Genocchi polynomials and derive the continued fraction
expansion for their generating function. 

\smallskip
Other aspects of Combinatorial Theory of Numbers are the
calculations of new {\it Seidel Triangles} [7] or the discovery of
new geometric structures for interpreting the
underlying calculus~[8]. 

\medskip
{\it   $2$. Combinatorial Study of Special Functions}. \quad The
problem is to build up combinatorial structures together
with their natural statistics, e.g., coloured derangements with
their cycle or descent numbers, that account for analytical
properties of the classical orthogonal polynomials. A thorough
study of nonnegativity properties of the {\it linearization
coefficients} had been solved
earlier by Zeng for the class of Sheffer polynomials. An
interesting joint study of the $q$-Laguerre polynomials and
their moments has been derived by him~[26]. In [14] a
combinatorial device is proposed for summing series of
products of orthogonal polynomials, essentially the Tchebychev
polynomials of the two kinds.

\medskip
{\it $3$. $q$-Eulerian Calculus}.\quad Following the
work of Denert on the calculation of zeta functions for
hereditary orders of some central simple algebras Han~[15] has
introduced a new family of {\it partially commutative monoids}
that enabled him to prove that the bivariate statistic (exc,den)
was Euler-mahonian. This work has been extended to the case
where the underlying alphabet is partitioned into two
subclasses of letters, the small letters and the large letters,
that lead to different classes of statistics (see [1, 2, 3, 4]).
This also provides a natural extension of the classical
statistical study of the symmetric group ${\cal S}_n$ towards
the analogous study of the group~$B_n$ of the {\it signed
permutations}.

\smallskip
When going from $A_n$ to $B_n$ (from a single alphabet to a
double-class alphabet) new tools are to be found or updated.
The so-called {\it MacMahon Verfahren} is such a tool; it has
been updated in~[11] and applied in~[3, 12, 13]. The study of
equidistribution properties has led to the discovery of
new relations, such as {\it bipartitional relations}~[12] that
need to be characterized or studied for their own sakes (see
[17, 18, 19]).

\goodbreak
\bigskip
{\it $4$. Analytical Tool Study}.\quad In section we mention the
study of polynomials defined by difference equations such as in
[28], or the analytical study of combinatorial polynomials, such
as the Eulerian polynomials, when the integers occurring in their
definitions are replaced by their $q$-analogues [29] or by
arbitrary complex numbers. The problem is to study under what
conditions the other formulas involving those polynomials are
preserved.

\bigskip
{\it $5$. Miscellaneous}.\quad
In this section are included the study of combinatorial
constructions derived in the theory of {\it symmetric
functions}. For instance, the construction by Kerov and his
collaborators to interpret the Foulkes polynomials still needs a
solid framework. A very natural geometric set-up to describe
such a construction is due to Han~[16].

\smallskip
Finally, new combinatorial structures are to be found to prove
the identities of Classical Invariant Theory, such as the
non-commutative extension of the {\it Capelli identity} found
recently by Howe. The young Slama is preparing a memoir on the
subject.

\smallskip
Not included in the following list, but already discussed in the
weekly Strasbourg seminar, the {\it Combinatorial Applications of
the resultant}, as proposed by Jouanolou,
will soon appear as a basic tool in Algebraic
Combinatorics.

\bigskip
{\bf C. Publications.}

\medskip
Let us cite the papers of the group that have published or been
accepted, or been submitted in 1994-95, under the EEC
Algebraic Programme.
%\parskip 3pt


\medskip
\article \nb|Robert J. Clarke and Dominique
Foata|Eulerian Calculus, I: univariable 
statistics|Europ. J. Combinatorics|15|1994|345--362|

\article \nb|Robert J. Clarke and Dominique
Foata|Eulerian Calculus, II: an  extension of Han's
fundamental transformation|Europ. J.
Combinatorics|16|1995|221--252|

\divers\nb|Robert J. Clarke and Dominique
Foata|Eulerian Calculus, III: the ubiquitous Cauchy formula,
{\sl Europ. J. Combinatorics}, to appear, {\oldstyle 1995}|

\divers \nb|Robert J. Clarke and Dominique
Foata|Eulerian Calculus, IV: specializations, {\sl S\'eminaire
Lotharingien de Combinatoire}, {\bf 32}b, {\oldstyle
1994}, 9pp|

\article \nb|Dominique Dumont|Conjectures
sur des sym\'etries ternaires li\'ees aux nombres de Genocchi
|Disc. Math.|139|1995|| 

\divers \nb|Dominique Dumont, A.
Ramamonjisoa|Grammaire de Ramanujan et Arbres de Cayley,
{\sl Electron. J. Math.}, {\oldstyle 1995} (to appear)|

\divers \nb|Dominique Dumont|Further
Triangles of Seidel-Arnold type and continued fractions related
to Euler and Springer numbers, {\sl Adv. in Applied Math.},
{\oldstyle 1995} (to appear)|

\article \nb|Dominique Dumont, Arthur
Randrianarivony|D\'erangements et nombres de Genocchi|Disc.
Math.|132|1994|37--49| 

\article \nb|Dominique Dumont and Arthur Randrianarivony|Sur une
extension des nombres de Genocchi|Europ. J.
Combinatorics|16|1995|147--151|

\article \nb|Dominique Dumont and Jean 
Zeng|Further results on the Euler and Genocchi numbers |Aequat.
Math.|47|1994|31--42| 

\divers \nb|Dominique Foata|Les distributions
Euler-Mahoniennes sur les mots, to appear in
{\sl Discrete Math.}, {\oldstyle 1995}|

\divers \nb|Dominique Foata and Doron Zeilberger|Graphical
Major Indices, to appear in 
{\sl J. of Computational and applied Math.}, {\oldstyle 1995}|

\divers \nb|Dominique Foata and Christian
Krattenthaler|Graphical Major Indices, II, 
{\sl S\'eminaire Lotharingien de Combinatoire}, {\bf 34}k,
Works in Progress, 16~pp|

 \divers\nb|Dominique Foata and Guoniu Han|Nombres
de Fibonacci et poly\-n\^omes orthogonaux, in {\sl Leonardo
Fibonacci: il tempo, le opere, l'ere\-dit\`a scientifica} [M.
Morelli and M. Tangheroni, eds.], p.~179--208. Pacini, Roma|

\article\nb|Guoniu Han|Une transformation fondamentale sur les
r\'ear\-ran\-ge\-ments de mots|Adv. in Math.|105|1994|26--41|

\divers\nb|Guoniu Han|Une version g\'eom\'etrique de la
construction de Kerov-Kirillov-Reshetikhin, {\sl S\'eminaire
Lotharingien de Combinatoire}, {\bf 31}, Publ. IRMA
Strasbourg, 1994/021, p.~71--85|

\divers\nb|Guoniu Han|The $k$-extension of a mahonian
statistic, to appear in {\sl Adv. in Appl. Math.}, {\oldstyle 1995}|

\divers \nb|Guoniu Han|Ordres bipartitionnaires et statistiques
sur les mots, to appear in {\sl Electronic J. Combinatorics},
{\oldstyle 1995}|

\divers\nb|Guoniu Han|Une d\'emonstration ``v\'erificative" d'un
r\'esultat de Foata-Zeilberger sur les relations
bipartitionnaires, to appear in 
{\sl J. of Computational and applied Math.}, {\oldstyle 1995}|

\divers\nb|Arthur Randrianarivony|Polyn\^omes de
Dumont-Foata g\'en\'eralis\'es, submitted for publication,
{\oldstyle 1995}|

\divers\nb|Arthur Randrianarivony|Fractions continues,
$q$-nombres de Catalan et $q$-polyn\^omes de Genocchi,
submitted for publication, {\oldstyle 1995}|

\divers\nb|Arthur Randrianarivony|$q,p$-analogues des nombres
de Catalan, submitted for publication, {\oldstyle 1995}|

\article \nb|Arthur Randrianarivony and Jean Zeng|Sur
une extension des nombres d'Euler et les records des
permutations alternantes|J. Combin. Th. Ser. A|68|1994|86-99|

\divers \nb|Arthur Randrianarivony and Jean Zeng|Une famille de
polyn\^omes qui interpole plusieurs suites classiques de
nombres, {\sl Actes du 31e S\'eminaire Lotharingien},  
{\oldstyle 1994}, p.~103-126, to appear in {\sl Adv. Appl. Math}|

\divers \nb|Arthur Randrianarivony and Jean Zeng|Some
equidistributed statistics on Genocchi  permutations, to appear
in {\sl Electronic J. Combin.},  {\oldstyle 1995}|

\divers \nb|Jean Zeng|The $q$-Stirling numbers, continued
fractions and the $q$-Charlier and $q$-Laguerre polynomials, 
{\sl J. of Computational and applied Math.}, {\oldstyle 1995}
(to appear)|

\divers \nb|Jean Zeng|Sur quelques propri\'et\'es de sym\'etrie des
nombres de Ge\-nocchi, {\sl Proc. of the Fifth conference on
FPSAC}, {\oldstyle 1993}, Florence, to appear in {\sl Disc. Math.}, 
{\oldstyle 1995}|

\divers \nb|Jean Zeng|Multinomial convolution polynomials, 
to appear in {\sl Disc. Math.}, {\oldstyle 1995}|

\article \nb|Jean Zeng and Changui Zhang|A $q$-analog of
Newton's series,  Stirling functions and Eulerian
functions|Results in Math.|25|1994|370-391|

\bigskip\bigskip
\line{Strasbourg, \hfil May 25, 1995}

\bigskip\bigskip

Dominique Foata,

Universit\'e Louis Pasteur, 

D\'epartement de math\'ematique et

Institut de Recherche Math\'ematique Avanc\'ee,

7, rue Ren\'e-Descartes,

F-67084 Strasbourg.

\medskip
Tel. : [33] 88 41 64 18

Fax : [33] 88 61 90 69

Email : foata@math.u-strasbg.fr

\bye






