\usepackage{mathtools}
\usepackage{mathrsfs}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{longtable}
\usepackage{ltcaption}
\usepackage{xcolor}
\usepackage{tikz}

\definecolor{axeZ}{RGB}{20,81,204}
\definecolor{axeY}{RGB}{212,43,11}
%\definecolor{axeX}{RGB}{136,189,30}
\definecolor{axeX}{rgb}{0.33, 0.42, 0.18}
\newcommand\perm[2]{\ensuremath{({\color{axeZ}#1}, {\color{axeY}#2})}}
\newcommand\bsig{{\ensuremath{\boldsymbol{\sigma}}}}
\newcommand\bi{{\ensuremath{\boldsymbol{i}}}}
\newcommand\bj{{\ensuremath{\boldsymbol{j}}}}
\newcommand\bbasig{{\ensuremath{\overline{\boldsymbol{\sigma}}}}}
\newcommand\basig{{\ensuremath{\overline{\sigma}}}}
\newcommand\bapi{{\ensuremath{\overline{\pi}}}}
\newcommand\bpi{{\boldsymbol{\pi}}}
\newcommand\btau{{\boldsymbol{\tau}}}
\DeclareMathOperator{\dir}{dir}
\newcommand\bdir{{\dir}}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\rev}{rev}
\DeclareMathOperator{\Sep}{Sep}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\Id}{Id}
\newcommand\new{\text{new}}
\newcommand\matSym[4]{\ensuremath{\big(\begin{smallmatrix}
			#1 & #2\\
			#3 & #4
		\end{smallmatrix}\big)}}
\newcommand\matSymT[9]{\ensuremath{\big(\begin{smallmatrix}
			#1 & #2 & #3\\
			#4 & #5 & #6 \\
			#7 & #8 & #9
		\end{smallmatrix}\big)}}

\newenvironment{centermath}
{\begin{center}$\displaystyle}
			{$\end{center}}


\newcommand\vinpat[2]{\ensuremath{#1|_{\color{axeX}#2}}}
\newcommand\vinpatb[3]{\ensuremath{#1|_{{\color{axeX}#2},{\color{axeY}#3}}}}
\newcommand\vinpatd[5]{\ensuremath{\vinpat{\perm{#1}{#2}}{{{\color{axeX}#3},{\color{axeY}#4},
					{\color{axeZ}#5}}}}}

\newcommand\baxpa{\ensuremath{\vinpatb{2413}{2}{2}}}
\newcommand\baxpb{\ensuremath{\vinpatd{312}{213}{1}{2}{.}}}
\newcommand\baxpc{\ensuremath{\vinpatd{3412}{1432}{2}{2}{.}}}
\newcommand\baxpd{\ensuremath{\vinpatd{2143}{1423}{2}{2}{.}}}
\newcommand\sbaxpa{\ensuremath{\Sym(\baxpa)}}
\newcommand\sbaxpb{\ensuremath{\Sym(\baxpb)}}
\newcommand\sbaxpc{\ensuremath{\Sym(\baxpc)}}
\newcommand\sbaxpd{\ensuremath{\Sym(\baxpd)}}



\newcommand\patt{\ensuremath{\Sym(\perm{132}{213})}}
\newcommand\TODO{{\color{red}TODO}}

%% First example
\newcommand\permfirst{\perm{253146}{654321}}
\newcommand\permfirstproj{{\color{axeX}364251}}
\newcommand\permbaxter{\perm{14386527}{47513268}}
