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\def\FoKrAA{P1}
\def\FuKrAA{P2}
\def\FuKrAB{P3}
\def\GeKrAA{P4}
\def\GuKrAA{P5}
\def\KratAY{P6}
\def\KratAZ{P7}
\def\KratBZ{P8}
\def\KratNZ{P9}
\def\KratBC{P10}
\def\KrPrAA{P11}


\def\SunaAD{R12}
\def\ProcAK{R11}
\def\MilnAG{R10}
\def\LitPAA{R9}
\def\HeTrAA{R8}
\def\GaRaAA{R7}
\def\FoZeAE{R6}
\def\CoHeAA{R5}
\def\ConcAC{R4}
\def\BurgAC{R3}
\def\AbKuAC{R2}
\def\AbhyAB{R1}

\rightheadtext{Algebraic combinatorics in Vienna}
\leftheadtext{Algebraic combinatorics in Vienna}

\head Algebraic combinatorics in Vienna \endhead
The researchers in Vienna are {\it Christian Krattenthaler, Markus
Fulmek}, and {\it Maria Prohaska}. The investigations mainly concentrated on the
combinatorics of tableaux and nonintersecting lattice paths and can
be roughly divided into four topics: \linebreak
(I)~Counting nonintersecting
lattice paths with a given total number of turns and the computation
of Hilbert series for determinantal and pfaffian rings,
(II)~Classical group characters, (III)~Tableau algorithms and
bijections, (IV)~Multiple basic hypergeometric series. Finally, in
another direction, research was undertaken in collaboration with the
Strasbourg group on (V)~Permutation statistics.

ad {\bf(I)~Counting nonintersecting
lattice paths with a given total number of turns and the computation
of Hilbert series for determinantal and pfaffian rings}. As shown by
\cite{\AbhyAB, \AbKuAC, \ConcAC, \CoHeAA, \HeTrAA}, the computation
of Hilbert series for determinantal and pfaffian rings boils down to 
counting families of nonintersecting lattice
paths with given starting and end points and a given total number of turns
in certain regions. Here the advances have been significant. All the
enumeration problems that come from determinantal rings, rings
generated by minors of symmetric matrices, one-sided ladder
determinantal rings, and pfaffian rings could be solved by developing
a combinatorial theory for ``non-crossing two-rowed arrays". See the
accompanying article \cite{\KratBZ} for more details. The
forth-coming papers \cite{\KratNZ, \KrPrAA} will contain a full
account of these results.

ad {\bf(II)~Classical group characters}. Here the main interest was
on the combinatorics of formulas for classical group characters. In
\cite{\FuKrAA} the Gessel--Viennot methodology of nonintersecting
lattice paths was extended to provide combinatorial proofs of all the
Jacobi--Trudi type determinant formulas for (irreducible) symplectic
and orthogonal characters, and the symplectic and odd orthogonal
Giambelli identities. Only the even Giambelli identity could not be
given a combinatorial proof. As a by-product, in \cite{\FuKrAB} for
the first time
Proctor's \cite{\ProcAK} and Sundaram's \cite{\SunaAD} odd orthogonal
tableaux, which are two very different combinatorial descriptions of
odd orthogonal characters, could be related to each other by a
bijection. Besides, classical combinatorial methods and tableau
descriptions coming from algebraic geometry \cite{\LitPAA} were
successfully combined in \cite{\KratBC} to obtain a number of new
identities for classical group characters.

ad {\bf(III)~Tableau algorithms and bijections}. Here, a new
algorithm, a ``modified jeu de taquin", was found and applied to
provide beautiful bijective proofs for the hook-content formula for
Schur functions \cite{\KratAY} and similar formulas for super Schur
functions \cite{\KratAZ}. 

ad {\bf(IV)~Multiple basic hypergeometric series}. 
Inspired by Burge \cite{\BurgAC}, 
in \cite{\GeKrAA} we introduce new objects into the theory of
partitions, ``cylindric partitions". Again, there is a close
relationship with nonintersecting lattice paths of a particular type. 
We extend almost all of Burge's results (which, in our language, concern
two-rowed cylindric partitions) to cylindric partitions with an
arbitrary number of rows.
It is then shown that cylindric partitions 
can be used to derive multiple basic hypergeometric
series associated with root systems. In particular, $\tilde
A_r$ basic hypergeometric summations of Milne \cite{\MilnAG} 
could be given new and
elementary proofs. Besides, several new $\tilde A_r$ basic
hypergeometric summation and transformation formulas were found.
Finally, in \cite{\GuKrAA} new $A_r$ extensions of Heine's
$_2\phi_1$-transformations \cite{\GaRaAA, (1.4.1),
(1.4.5), (1.4.6)} were found.

ad {\bf(V)~Permutation statistics}. 
In \cite{\FoKrAA} generalizations of the classical statistics ``maj" and
``inv" (the major index and the number of inversions) on words are
introduced that depend on a graph on the underlying alphabet and the
behaviour of each letter at the end of a word. The question of
characterizing those graphs that lead to equidistributed ``maj" and
``inv" is then posed and answered. This work extends a previous result of
Foata and Zeilberger \cite{\FoZeAE} who considered the same problem under the
assumption that all letters have the same behaviour at the end of a
word.
\Refs\nofrills{Papers}
\ref\no \FoKrAA\by D.    Foata and C. Krattenthaler \paper Graphical
major indices II\paperinfo in preparation \jour \vol \pages \endref

\ref\no \FuKrAA\by M.    Fulmek and C. Krattenthaler 
\paper Lattice path proofs for determinant formulas for symplectic and orthogonal characters
\paperinfo preprint\endref

\ref\no \FuKrAB\by M.    Fulmek and C. Krattenthaler 
\paper Bijections between odd orthogonal tableaux
\paperinfo preprint\endref


\ref\no \GeKrAA\by I. M. Gessel and C. Krattenthaler \yr 19?? 
\paper Cylindric partitions\paperinfo preprint
\jour \vol 
\pages \endref

\ref\no \GuKrAA\by R. A. Gustafson and C. Krattenthaler
\paper Heine transformations for a new kind of basic hypergeometric series in $U(n)$
\paperinfo preprint\endref


\ref\no \KratAY\by C.    Krattenthaler \yr 19?? 
\paper An involution principle-free bijective proof of Stanley's hook-content formula
\paperinfo preprint\jour \vol 
\pages \endref

\ref\no \KratAZ\by C.    Krattenthaler \yr 19?? 
\paper A bijective proof of the hook-content formula for super Schur functions and a modified jeu de taquin 
\paperinfo preprint \endref

\ref\no \KratBZ\by C. Krattenthaler \yr 1995 \paper Counting 
nonintersecting lattice paths with turns \jour Seminaire
Lotharingien, item ``Back Issues"\vol 34\pages ibid\endref

\ref\no \KratNZ\by C.    Krattenthaler \paper Non-crossing two-rowed arrays
\jour in preparation\endref  

\ref\no \KratBC\by C.    Krattenthaler \yr 19 \paper Classical group characters of
rectangular and nearly rectangular shape
\paperinfo in preparation\jour \vol \pages \endref

\ref\no \KrPrAA\by C.    Krattenthaler and M. Prohaska \yr 19?? 
\paper A remarkable formula for counting nonintersecting lattice paths in a ladder with respect to turns
\paperinfo in preparation\jour \vol 
\pages \endref


\endRefs



\Refs

\ref\no \AbhyAB\by S. S. Abhyankar \yr 1988 
\book Enumerative combinatorics of Young tableaux
\publ Marcel Dekker
\publaddr New York, Basel\endref

\ref\no \AbKuAC\by S. S. Abhyankar and D. M. Kulkarni \yr 1989 
\paper On Hilbertian ideals
\jour Linear Alg\. Appl\.\vol 116
\pages 53--76\endref

\ref\no \BurgAC\by W. H. Burge \yr 1993 
\paper Restricted partition pairs
\jour J. Combin\. Theory Ser.~A\vol 63
\pages 210--222\endref

\ref\no \ConcAC\by A.    Conca \yr 1994 
\paper Symmetric ladders
\jour Nagoya Math\. J.\vol 136
\pages 35--56\endref

\ref\no \CoHeAA\by A.    Conca and J. Herzog \yr 1994 
\paper On the Hilbert function of determinantal rings and their canonical module 
\jour Proc\. Amer\. Math\. Soc\. \vol 122 
\pages 677--681\endref

\ref\no \FoZeAE\by D.    Foata and D. Zeilberger \yr 19?? \paper Graphical 
major indices\paperinfo preprint\jour \vol \pages \endref

\ref\no \GaRaAA\by G.    Gasper and M. Rahman \yr 1990 
\book Basic hypergeometric series
\publ Encyclopedia of Mathematics And Its Applications~35, Cambridge University Press
\publaddr Cambridge\endref

\ref\no \HeTrAA\by J.    Herzog and N. V. Trung \yr 1992 
\paper Gr\"obner bases and multiplicity of determinantal and Pfaffian ideals
\jour Adv\. in Math\.\vol 96
\pages 1--37\endref

\ref\no \LitPAA\by P.    Littelmann \yr 1990 
\paper A generalization of the Littlewood--Richardson rule
\jour J. Algebra\vol 130
\pages 328--368\endref

\ref\no \MilnAG\by S. C. Milne \yr 1992 \paper Classical partition functions 
and the $U(n+1)$ Rogers--Selberg identity\jour 
Discrete Math\.\vol 99\pages 199---246\endref

\ref\no \ProcAK\by R. A. Proctor \yr 1994 
\paper Young tableaux, Gelfand patterns, and branching rules for classical groups 
\jour J.~Algebra \vol 164 
\pages 299--360\endref

\ref\no \SunaAD\by S.    Sundaram \yr 1990 
\paper Orthogonal tableaux and an insertion algorithm for $SO(2n+1)$
\jour J. Combin\. Theory Ser.~A\vol 53
\pages 239--256\endref

\endRefs
\end

