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\input math-stuff

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%  Spezifische Macros:                                              %
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\def\n{N}		% Number of variables 

\def\indexthis#1#2{{#1}^{(#2)}}
\def\sapadd#1#2#3{{#1}\!+^{#2}_{#3}}
\def\sappeel#1{\left.#1\right\downarrow}
\def\sappeeldown#1#2{\left.#1\right\downarrow^{#2}}
\def\sappeelup#1#2{\left.#1\right\uparrow_{(#2)}}

\def\cop{{\sl\bf cp}}
\def\coc{{\sl\bf rop}}
\def\nop{{\sl\bf npm}}
\def\pointconf{\EM{{\bf c}onfiguration of {\bf p}oints} (short: \cop){\gdef\pointconf{\cop}}}
\def\pointarr{\EM{{\bf r}adial {\bf o}rientation of {\bf p}oints} (short: \coc){\gdef\pointarr{\coc}}}
\def\pointmatch{\EM{{\bf n}oncrossing {\bf p}erfect {\bf m}atching} (short: \nop){\gdef\pointmatch{\nop}}}
\def\nilp{non\-in\-ter\-sec\-ting lat\-ti\-ce path}

% Notation for determinants
\def\mdetof#1{\left\vert#1\right\vert}
% Matrices and minors
\def\detwhole{{\mathbf m}}
\def\detminor#1#2{\detwhole_{\pas{#1},\pas{#2}}}
\def\detspecminor#1#2#3{\pas{#1}_{#2,#3}}
\def\detcominor#1#2{{\overline{\detwhole}}^{\pas{#1},\pas{#2}}}
\def\jtmat#1{{\mathbf h}_{#1}} % Jacobi-Trudi-Matrix
% Minor obtained by deleting rows and/or columns:
\def\deleterowcols#1#2#3{\brk{#1}_{\underline{#2}\vert\underline{#3}}}
\def\deleterowcols#1#2#3{\brk{#1}_{\text{\sout{$#2$}}\vert\text{\sout{$#3$}}}}
\def\deleterow#1#2{\deleterowcols{#1}{#2}{\hphantom{0}}}
\def\deletecol#1#2{\deleterowcols{#1}{\hphantom{0}}{#2}}
% Range of numbers
\def\range#1{\brk{#1}}
\def\shiftrange#1#2{\pas{\range#1+#2}}
% Sum of elements in a set of integers
\def\sumset#1{{\scriptstyle\pas{\sum_{x\in #1}x}}}
\def\sumset#1#2{\Sigma_{#1\subseteq#2}}
% Exchange subsets
\def\shuffle#1#2#3{\pas{{\scriptscriptstyle #2\!\leftarrow}\!#1\!{\scriptscriptstyle\leftarrow \!#3}}}
\def\shuffle#1#2#3{\pas{\left.#1\right\vert^{{\scriptscriptstyle#2\rightarrow}}_{{\scriptscriptstyle\leftarrow#3}}}}
% Minors obtained by selecting subsets of rows and columns
\def\minor#1#2#3{\pas{#1}_{#2,\;#3}}
% Minor obtained by deleting rows and/or columns:
\def\deleterowcols#1#2#3{\minor{#1}{\overline{#2}}{\overline{#3}}}
\def\deleterow#1#2{\deleterowcols{#1}{\setof{#2}}{\emptyset}}
\def\deletecol#1#2{\deleterowcols{#1}{\emptyset}{\setof{#2}}}
% Product of complementary minors:
\def\prodcminors#1#2#3{\minor{#1}{#2}{#3}\cdot\deleterowcols{#1}{#2}{#3}}
\def\detprodcminors#1#2#3{\detof{\minor{#1}{#2}{#3}}\cdot\detof{\deleterowcols{#1}{#2}{#3}}}
% Limit of a partition
\def\limes#1{#1_\infty}
% Constant partition
% \def\allconstpart#1{\pas{{#1}^{\infty}}}
\def\allconstpart#1{\pas{\overline{#1}}}
\def\periodic#1{\dot{#1}}
% Partition of length #2 with constant part #1
%\def\constpart#1#2{\pas{{#1}^{\llbracket{#2}\rrbracket}}}
\def\constpart#1#2{\pas{{#1}^{\lfloor{#2}\rceil}}}
% Shape
\def\shape#1#2{#1/#2}
% Weight of partition/shape
\def\partweightof#1{\cardof{#1}}
% Partition obtained by deleting k-th part
% \def\deletepart#1#2{\operatorname{del}_{#2}\pas{#1}}
%\def\deletepart#1#2{{#1}^{\underline{#2}}}
%\def\deletepart#1#2{\pas{#1}^{\text{\sout{$#2$}}}}
\def\deletepart#1#2{{#1}^{\overline{#2}}}
% Partition obtained by adding k to all parts
\def\addtoparts#1#2{\pas{#1+\constpart{#2}{m}}}
\def\subtoparts#1#2{\pas{#1-#2}}
% Horizontal and vertical arcs starting in some point
\def\harc#1{{\scriptstyle #1}^\to}
\def\harc#1{\overline{#1}}
\def\harc#1{a^h_{#1}}
\def\varc#1{{\scriptstyle #1}^\uparrow}
\def\varc#1{{#1}\vert}
\def\varc#1{a^v_{#1}}
% Number of entries k in tableau T:
\def\nofentries#1#2{\#\of{#1,#2}}
% Generating function
\def\gfof#1{{\mathbf{gf}\of{#1}}}
% Set of lattice paths
\def\paths#1#2{{\mathfrak P}\of{#1\to#2}}
\def\pathsmod#1#2#3{{\mathfrak P}^{#3}\of{#1\to#2}}
% Generating function of paths
\def\gfpaths#1#2{\gfof{\paths{#1}{#2}}}
% Identity permutation
\def\idperm{\operatorname{id}}
% All terms in the Jacobi-Trudi-Determinant
\def\alljt{{\mathfrak D}}
% Intersecting terms in the Jacobi-Trudi-Determinant
\def\ijt{{\mathfrak I}}
% Nonintersecting terms in the Jacobi-Trudi-Determinant
\def\nijt{{\mathfrak N}}
% Involution
\def\involution#1{{\mathbf{#1}}}
\def\invol{\involution{i}}
% Indicate vector
\def\asvector#1{{\mathbf #1}}
% Subpartition
\def\subpart{\trianglelefteq}
% Identity-Style
\def\identity#1{{\mathcal #1}}
% Cop/Nop/Rop-Style
\def\cnr#1{{\mathbf #1}}
% Arc in some directed graph
\def\arca{\alpha}
% Generic matrix
\def\mata{a}
%We define certain alternating sums of such products by enclosing the members
%of some fixed subset $A$ of entries in such tableau by
\def\symmfixed#1{\overline{\symm_{#1}}}

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% Referee's points:
\def\lgti{weight-preserving bijection between Young tableaux and \nilp s}
\def\gedankenstrich{---}
\def\bindestrich {-}

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\def\tbsum#1{\brk{#1}}
\def\tbsum#1{T^{A\cup B}_{\circ\equiv #1}}
\def\tbsub#1{T^{A\cup B}_{\circ\supseteq #1}}
\def\tbbsum#1{\brk{#1}}

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\begin{document}

\bibliographystyle{plain}

\title{Viewing determinants as nonintersecting lattice paths yields
	classical determinantal identities bijectively}

\author{Markus Fulmek\thanks{
Research supported by the National Research Network ``Analytic
Combinatorics and Probabilistic Number Theory'', funded by the
Austrian Science Foundation. 
} \\
\small Fakult\"at f\"ur Mathematik \\[-0.8ex]
\small Nordbergstra\ss e 15, A-1090 Wien, Austria \\
\small\tt Markus.Fulmek@univie.ac.at
}
%\address{Fakult\"at f\"ur Mathematik, % \\ Universit\"at Wien\\
%Nordbergstra\ss e 15, A-1090 Wien, Austria}
%\email{{\tt Markus.Fulmek@Univie.Ac.At}\newline\leavevmode\indent
%{\it WWW}: {\tt http://www.mat.univie.ac.at/\~{}mfulmek}
%}

%\date{\today}
%\thanks{
%Research supported by the National Research Network ``Analytic
%Combinatorics and Probabilistic Number Theory'', funded by the
%Austrian Science Foundation. 
%}

\date{\dateline{Feb 16, 2012}{Jul 12, 2012}{Aug 16, 2012}\\
\small Mathematics Subject Classifications: 05A19, 05E05}

\maketitle

\begin{abstract}
In this paper, we show how general determinants may be \EM{viewed} as generating functions of
nonintersecting lattice paths, using the \lgt\bindestrich method
% interpretation of semistandard Young tableaux
and the Jacobi\bindestrich  Trudi identity together with elementary observations.
After some preparations, this point of view provides ``graphical proofs''
for classical determinantal identities like the Cauchy\bindestrich Binet formula, Dodgson's
condensation formula, the Pl\"ucker relations, Laplace's expansion and Turnbull's identity.
Also, a determinantal identity generalizing Dodgson's condensation formula is presented,
which might be new.
\end{abstract}


%\section{Additional remarks}
%\input newstuff
%
\section{Introduction}\label{sec:intro}
% {\red Endpunkte von rechts oder von links gezŠhlt??? Im Text klar machen!!!}
\input intro

\section{Basic definitions}\label{sec:background}
\input background

\section{Determinants as nonintersecting lattice paths}\label{sec:main}
\input main

\section{Products of determinants/Overlays of lattice paths}\label{sec:overlays}
\input overlays

\section{Applications: Classical determinantal identities}\label{sec:apps}
\input apps

\bibliography{paper}

\end{document}
