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\comment
\topmatter 
\title THE MAJOR COUNTING OF NONINTERSECTING LATTICE PATHS AND 
GENERATING FUNCTIONS FOR TABLEAUX\endtitle 
\author C.~Krattenthaler
\endauthor 
\affil 
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.
\endaffil 
\address Institut f\"ur Mathematik der Universit\"at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
\endaddress
\email KRATT\@Pap.Univie.Ac.At (In), KRATT\@AWIRAP (Bitnet)\endemail
%\dedicatory \enddedicatory
%\date Preliminary Version\enddate
%\thanks \endthanks
\subjclass Primary 05A15;
 Secondary 05A10, 05A17, 05A30, 05E10, 33D20
\endsubjclass
\keywords Nonintersecting lattice paths, major index, tableaux,
symmetric plane partitions,
generating functions, Robinson--Schensted--Knuth correspondence, 
basic hypergeometric series in $U(n)$ and $Sp(n)$, basic 
hypergeometric series for $A_l$ and $C_l$\endkeywords
\abstract A theory of counting nonintersecting lattice paths by the
major index and generalizations of it is developed. We obtain determinantal
expressions for the corresponding generating functions for families of
nonintersecting lattice paths with given starting points and given
final points, where the starting points lie on a line parallel to
$x+y=0$. In some cases these determinants can be evaluated to result
in simple products. As applications we compute the generating
function for tableaux with $p$ odd rows, with at most $c$ columns,
and with parts between $1$ and $n$.
Besides, we compute the generating function for the same kind of
tableaux which in addition have only odd parts. We thus also obtain a
closed form for the generating function for symmetric plane
partitions with at most $n$ rows, with parts between $1$ and $c$, and
with $p$ odd entries on the main diagonal. In each case the
result is a simple product. By summing with respect to $p$ we provide
new proofs of the Bender--Knuth and MacMahon (ex-)Conjectures, which
were first proved
by Andrews, Gordon, and Macdonald. The link between nonintersecting
lattice paths and tableaux is given by variations of the Knuth
correspondence.
\endabstract

\toc
\title \rm{\bf I. Introduction} \leaderfill {\bf 2}\endtitle
\title \rm{\bf II. Definitions and Preliminaries} \leaderfill {\bf
5}\endtitle
\head {2.1\hss} \leavevmode\hskip0.4cm Definitions and Notation \leaderfill
5\endhead
\head {2.2\hss} \leavevmode\hskip0.4cm Two involutions and
nonintersecting lattice paths \leaderfill 8\endhead

\title \rm{\bf III. Counting by Major Index} \leaderfill {\bf 14}\endtitle
\head {3.1\hss} \leavevmode\hskip0.4cm Without restrictions \leaderfill
14\endhead
\head {3.2\hss} \leavevmode\hskip0.4cm With a diagonal boundary \leaderfill
17\endhead
\head {3.3\hss} \leavevmode\hskip0.4cm Tableaux generating functions \leaderfill
27\endhead
\title \rm{\bf IV. Counting by Strange Major Index} \leaderfill {\bf
34}\endtitle
\head {4.1\hss} \leavevmode\hskip0.4cm Without restrictions \leaderfill
35\endhead
\head {4.2\hss} \leavevmode\hskip0.4cm With a diagonal boundary \leaderfill
38\endhead
\head {4.3\hss} \leavevmode\hskip0.4cm Tableaux generating functions \leaderfill
51\endhead
\title \rm{\bf V. Detailed Proofs and Auxiliary Results} \leaderfill
{\bf 59}\endtitle
\head {5.1\hss} \leavevmode\hskip0.4cm The major index and strange
major index generating functions for lattice paths without
restrictions \leaderfill 59\endhead
\head {5.2\hss} \leavevmode\hskip0.4cm A correspondence for lattice
paths which cross the line $x=y$\linebreak\hphantom{g} \leaderfill 63\endhead
\head {5.3\hss} \leavevmode\hskip0.4cm A correspondence for pairs of
intersecting lattice paths \leaderfill 74\endhead
\head {5.4\hss} \leavevmode\hskip0.4cm Correspondences between tableaux and nonintersecting lattice
paths \leaderfill 88\endhead
\head {5.5\hss} \leavevmode\hskip0.4cm A determinant lemma \leaderfill
100\endhead
\head {5.6\hss} \leavevmode\hskip0.4cm A multiple basic hypergeometric summation \leaderfill
102\endhead
\title \rm{\bf References} \leaderfill {\bf 109}\endtitle
\endtoc
\endtopmatter
\endcomment

\document
\input maj12

\newpage

\input maj3

\newpage

\input maj4

\newpage

\input maj51
\input maj52

\newpage

\Refs
\ref\no1 \by G.~E.~Andrews\book The Theory of Partitions\publ
Addison--Wesley\publaddr Reading\yr 1976\bookinfo Encyclopedia of
Mathematics and its Applications, Vol.~2\endref
\ref\no2 \by G. E. Andrews\paper Plane partitions~I: The MacMahon
conjecture\inbook Studies in foundations and combinatorics\publ 
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\endRefs




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