Diagram of Greene Cells

I am working with coplactic diagrams of Greene cells and I need to make some examples and I can't figure out a way to draw the following diagram. Does anyone know the best way to make such a diagram? With tikz-cd one can use a fairly simple syntax.

\documentclass{article}
\usepackage{tikz-cd}

\begin{document}

$\begin{tikzcd}[arrows=dash,every label=\textstyle] 3214 \arrow[r,"3"] \arrow[d,"3"] & 4213 \arrow[r,"2"] \arrow[d,"3"] & 4312 \arrow[d,"2"] \\ 3241 \arrow[r,"3"] \arrow[d,"2"] & 4231 \arrow[r,"1"] \arrow[d,"1"] & 4132 \arrow[d,"1"] \\ 3421 \arrow[r,"2"] & 2431 \arrow[r,"1"] & 1432 \end{tikzcd}$

\end{document} • +1 I was oscillating between both possibilities. – user121799 Apr 30 '18 at 21:01

This is one possibility. Other possibilities include working with edges and so on, but using copy and paste was faster.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}
\matrix (perms) [matrix of nodes,row sep=1cm,column sep=1cm]
{
3214 & 4213 & 4312 \\
3241 & 4231 & 4132 \\
3421 & 2431 & 1432 \\
};
\draw (perms-1-1) -- (perms-1-2) node[midway,above]{3};
\draw (perms-1-2) -- (perms-1-3) node[midway,above]{2};
\draw (perms-2-1) -- (perms-2-2) node[midway,above]{3};
\draw (perms-2-2) -- (perms-2-3) node[midway,above]{1};
\draw (perms-3-1) -- (perms-3-2) node[midway,above]{2};
\draw (perms-3-2) -- (perms-3-3) node[midway,above]{1};
%
\draw (perms-1-1) -- (perms-2-1) node[midway,right]{3};
\draw (perms-1-2) -- (perms-2-2) node[midway,right]{3};
\draw (perms-1-3) -- (perms-2-3) node[midway,right]{2};
\draw (perms-2-1) -- (perms-3-1) node[midway,right]{2};
\draw (perms-2-2) -- (perms-3-2) node[midway,right]{1};
\draw (perms-2-3) -- (perms-3-3) node[midway,right]{1};
\end{tikzpicture}
\end{document} Another solution, the result is similar.

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\node(SW) at (0,0){3421};
\node(SC) at (2,0){2431};
\node(SE) at (4,0){1432};
\node(CW) at (0,2){3241};
\node(CC) at (2,2){4231};
\node(CE) at (4,2){4132};
\node(NW) at (0,4){3214};
\node(NC) at (2,4){4213};
\node(NE) at (4,4){4312};
\draw(SW) -- node[above]{2} (SC) --  node[above]{1} (SE);
\draw(CW) --  node[above]{3} (CC) --  node[above]{1} (CE);
\draw(NW) --  node[above]{3} (NC) --  node[above]{2} (NE);
\draw(SW) --  node[right]{2} (CW) --  node[right]{3} (NW);
\draw(SC) --  node[right]{1} (CC) --  node[right]{3} (NC);
\draw(SE) --  node[right]{1} (CE) --  node[right]{2} (NE);
\end{tikzpicture}
\end{document}

It has nothing more wrt the previous one. Simply I started to write it before @marmot posted his answer. • well, even egreg posted his answer before mine. ;) – gvgramazio Apr 30 '18 at 20:33