%This is an AmS-TeX file %for 3 pages \input amstex \input amsppt.sty \NoBlackBoxes \magnification1200 \hsize15.6truecm \vsize22.8truecm \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