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% Theorem-environments                                              % 
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\newif\ifenglish
\englishtrue

% Italic body:
\newtheorem{ex}{\ifenglish Example\else Beispiel\fi}
\newtheorem{ax}{\ifenglish Axiom\else Prinzip\fi}
\newtheorem{numex}{\ifenglish Numerical Example\else Rechenbeispiel\fi}
\newtheorem{thm}{\ifenglish Theorem\else Satz\fi}
\newtheorem{pro}{Proposition}
\newtheorem{lem}{Lemma}
\newtheorem{rem}{\ifenglish Remark\else Bemerkung\fi}
\newtheorem{dfn}{Definition}
\newtheorem{cor}{\ifenglish Corollary\else Korollar\fi}

% Roman body:
% \theoremstyle{remark}
\newtheorem{obs}{\ifenglish Observation\else Beobachtung\fi}

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%  Spezielle Makros                                                 % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\figref#1{Figure~\ref{#1}}
\def\secref#1{Section~\ref{#1}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Abkuerzungen:                                                     % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\defeq{:=}
\def\wrongeq{\stackrel{=}{?}}
\def\absof#1{\left|#1\right|}
\def\setof#1{\left\{#1\right\}}
\def\of#1{\!\left(#1\right)}
\def\subof#1{_{\left(#1\right)}}
\def\pas#1{\left(#1\right)}
\def\brk#1{\left[#1\right]}
\def\sgn{{\operatorname{sgn}}} 
\def\var{{\operatorname{var}}}
\def\cov{{\operatorname{cov}}}
\def\corr{{\operatorname{corr}}}
\def\ip#1#2{{\left\langle #1,#2\right\rangle}}
\def\floor#1{{\left\lfloor #1\right\rfloor}}
\def\ceil#1{{\left\lceil #1\right\rceil}}
\def\bit{\begin{itemize}}
\def\eit{\end{itemize}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\ffact#1#2{\pas{#1}_{#2}}
\def\rfact#1#2{\pas{#1}^{#2}}

\def\C{{\mathbb C}}
\def\E{{\mathbb E}}
\def\H{{\mathbb H}}
\def\N{{\mathbb N}}
\def\P{{\mathbb P}}
\def\Q{{\mathbb Q}}
\def\R{{\mathbb R}}
\def\Z{{\mathbb Z}}

\def\i{{\mathbf i}}
\def\d{{\mathrm d}}
\def\e{{\mathbf e}}
\def\1{{\mathbf 1}}

\def\DF#1#2{D_{#1,#2}}	% Diskontfaktoren
\def\normdist{\mathcal N}

\def\PP{\mathbb{P}}
\def\dd{{\rm d}}

\newenvironment{LI}[1]%
{\begin{list}{#1}{\parsep0em \itemsep0.3em \labelwidth1em
\labelsep0.5em \leftmargin1.5em }} {\end{list}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Specific macros:                                                  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\lgt{Lindstr\"om--Gessel--Viennot}
\def\polylog#1#2{\operatorname{Li}_{#1}\of{#2}}
\def\detof#1{\left\vert{#1}\right\vert}
\def\esf{{\mathrm s}}
\def\ddd{m}	% Number/generating function of restricted Dyck paths
\def\gfddd{{\mathbf m}}	% Number/generating function of restricted Dyck paths
\def\OO{O}	% Big O
\def\oo{o}	% Small O
\def\ndivs#1{{\mathrm d}\of{#1}}	% Number of divisors
\def\pochhammer#1#2{\pas{#1}_{#2}}
%\def\fallfac#1#2{\pochhammer{#1 - #2 + 1}{#2}}
\def\fallfac#1#2{\left(#1\right)_{\left.#2\right\downarrow}}
\def\HH{{\mathcal H}}
\def\JJ{{\mathcal J}}
\def\PP{{\mathcal P}}
\def\mytheta{{\bar\vartheta}}
\def\HYPF#1#2#3#4#5{{}_{#1} F_{#2}\left[\begin{matrix}#3\\#4\end{matrix};#5\right]}
\def\singlesum{{\mathbf S_1}}
\def\doublesum{{\mathbf S_2}}
\def\dee{{\mathrm d}}
\def\deeof#1{{\mathrm d} #1}
\def\differ#1#2#3{\frac{\dee^{#3}}{\dee #2^{#3}}#1}
\def\Iverson#1{\brk{#1}}
\def\remark#1{\par\noindent{\sc Remark: #1}\par}
\def\path{{%\mathrm 
P}}
\def\figref#1{Figure~\ref{#1}}
\def\dummy#1{n_{#1}}
\def\minheight{\underline{m}}
\def\bigpas#1{\Bigl(#1\Bigr)}
\def\bdetof#1{\Bigl|#1\Bigr|}
\def\bbdetof#1{\Biggl|#1\Biggr|}


\def\pochhammer#1#2{\pas{#1}^{\overline{#2}}}
\def\fallfac#1#2{\pas{#1}^{\underline{#2}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Symmetrische Funktionen:                                          % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\symfs{\Lambda}	% subring of symmetric polynomials
\def\csymf{h}		% complete symmetric functions
\def\esymf{e}		% homogeneous symmetric functions

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Aktuelle Makros:                                                  % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\pfaffian{\operatorname{Pf}}
\def\hafnian{{\operatorname{Hf}}}
\def\per{\operatorname{per}}
\def\inv{\operatorname{inv}}
%
\def\matchings#1{{\mathcal M}_{#1}}
\def\superpositions#1{{\mathcal S}_{#1}}
\def\nofmatchings#1{M_{#1}}
\def\subgraph#1#2{\brk{#1-#2}}
\def\subgraphedges#1#2#3{\brk{#1-#2-#3}}
% \def\subgraph#1#2{G_{{\scriptscriptstyle-#2}}}
\def\weight{\omega}
\def\vertices{{\mathrm V}}
\def\edges{{\mathrm E}}
\def\orientation{\xi}
\def\symbolf{f\!}
\def\interlace{{\scriptstyle\rightleftarrows}}
\def\concat{{\scriptstyle\circ}}
\def\word#1{\pas{#1}}
\def\complement#1{\overline{#1}}
\def\without{\!\setminus\!}
\def\symdef#1#2{#1\,{\vartriangle}\,#2} % symmetric difference
\def\numof#1{\#\of{#1}}                % number of a set
\def\cardof#1{\left\vert{#1}\right\vert}                % cardinality of a set
\def\smatch{\mu}                        % single matching
\def\bicoloured#1#2#3{#1_{{ #2}\vert{ #3}}}
\def\theblues{{\blue\mathbf{b}}}
\def\thereds{{\red\mathbf{r}}}
\def\thecoloureds{{\black\mathbf{c}}}
\def\thewhites{{\black\mathbf{w}}}
\def\swapcolours#1{\chi_{#1}}
% \def\setsum#1{{\scriptstyle\Sigma}#1}
\def\setsum#1{{{\mathbf\Sigma}\of{\!#1}}}
%\def\SETSUM#1#2{{{\mathbf\Sigma}_{#2}\of{\!#1}}}
\def\SETSUM#1#2{{{\mathbf\Sigma}\of{{#1\subseteq#2}}}}
\def\symm{\mathfrak{S}}
\def\descof#1{d\,\of{#1}}
\def\LHS{left--hand side}
\def\RHS{right--hand side}
\def\symmeven#1{{\symm^{0}_{#1}}}
\def\step#1{Step #1: }
\def\semibip#1#2{S_{#1,#2}}
\def\completeG#1{K_{#1}}
\def\completebip#1#2{K_{#1,#2}}
\def\word#1{{\mathbf #1}}
\def\pmatch{matching}
\def\Pmatch{Matching}
%\def\strangeset#1#2{{\mathcal F}\of{#1, #2, \thecoloureds}}
%\def\strangeset{{\mathcal F}\of{\therights, \thelefts, \thewhites\|X}}
\def\strangeset{{\mathcal F}\of{G\|X}}
\def\markextra#1{#1}
\def\disjuni{\,\dot{\cup}\,}
\def\bb{bb}
\def\rb{rb}
\def\br{br}
\def\rr{rr}
\def\and{\wedge}
\def\indexed#1#2{{{\mathbf #1}}_{#2}}
\def\redindexed#1{\setof{r: #1}}
\def\blueindexed#1{\setof{b: #1}}
\def\edgeindexed#1{\setof{e: #1}}
\def\complement#1{\overline{#1}}
%\def\thelefts{{\gray\bar{\mathbf R}}}
%\def\therights{{\gray\bar{\mathbf B}}}
\def\thelefts{{\red{\mathbf R}}}
\def\therights{{\blue{\mathbf B}}}
\def\myrange#1{\setof{1,\dots,k}}
\def\som{superposition of matchings}
\def\soms{superpositions of matchings}

\def\schurf{s}
\def\length#1{\ell\of{#1}}
\def\tableau{T}


