How to improve the following codes to get the following diagram in Latex?

Question

I want to get the above diagram in Latex, but I don't know how to edit it. Any help will be appreciated. I post my effort in the following way.

\documentclass{article}
\usepackage[all]{xy}
\begin{document}
\[
\xymatrix@C1.5em{
&& \binom{\alpha_2}{0} \ar[dr]  && \ar[l] \binom{\alpha_2}{1}  \ar[dr] && 
\ar[l] \binom{\alpha_2}{2}   \\
&  \ar@{.>}[ur]\binom{\alpha_1}{0}  && \ar[l] \binom{\alpha_1}{1} \ar[ur]   
&& \ar[l] \binom{\alpha_1}{2} \ar@{.>}[ur]   \\
}
\]
\end{document}

Answer 1

Since xy package is not mandatory for you, I suggest a solution with a tikz matrix.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{matrix}
\tikzset{%
    mypoint/.style={circle, fill=black},
    middlearrow/.style={% this code is from https://tex.stackexchange.com/a/39283/101651
        decoration={markings,
            mark= at position 0.5 with {\arrow[scale=2]{#1}} ,
        },
        postaction={decorate}
    }
}

\begin{document}
    \begin{tikzpicture}
    \matrix[column sep=2.5em, row sep=8ex, inner sep=0pt, minimum width=4pt] {%
        %first row
        \node (startrow1) {}; &&
        \node[mypoint, label={[above,yshift=2pt]$\beta_{1}$}] (B1) {}; &&
        \node[mypoint, label={[above,yshift=2pt]$\beta_{2}$}] (B2) {}; && 
        \node[mypoint, label={[above,yshift=2pt]$\beta_{3}$}] (B3) {}; && 
        \node[text width=3em, align=center] (endrow1) {$\beta$};
        \\
        %second row
        \node (startrow2) {}; &
        \node[mypoint, label={[below,yshift=-8pt]$\alpha_{1}$}] (A1) {}; &&
        \node[mypoint, label={[below,yshift=-8pt]$\alpha_{2}$}] (A2) {}; && 
        \node[mypoint, label={[below,yshift=-8pt]$\alpha_{3}$}] (A3) {}; &&& 
        \node[text width=3em, align=center] (endrow2) {$\alpha$};
        \\
    };
    \draw[dashed] (startrow1) -- (B1)
    (B3) -- (endrow1)
    (startrow2) -- (A1)
    (A3) -- (endrow2);
    \draw[middlearrow={<}] (B1) -- (B2);
    \draw[middlearrow={<}] (B2) -- (B3);
    \draw[middlearrow={<}] (A1) -- (A2);
    \draw[middlearrow={<}] (A2) -- (A3);
    \draw[middlearrow={>}, dashed] (A1) -- (B1);
    \draw[middlearrow={>}] (B1) -- (A2);
    \draw[middlearrow={>}] (A2) -- (B2);
    \draw[middlearrow={>}] (B2) -- (A3);
    \draw[middlearrow={>}, dashed] (A3) -- (B3);
    \end{tikzpicture} 
\end{document}

Answer 2

another proposition with use of tikz package ... with use of \usetikzlibrary{arrows.meta, decorations.markings, positioning}:

documentclass[tikz, margin=3mm]{standalone}
\usetikzlibrary{arrows.meta, decorations.markings, positioning}

\begin{document}
    \begin{tikzpicture}[
node distance = 22mm and 11mm,
   dot/.style = {circle, fill=black, inner sep=0pt, minimum size=2mm,
                 node contents={}},
    ma/.style = {decoration={% middle arrows
                 markings, 
                 mark= at position 0.5 with {\arrow{Straight Barb[length=3pt]}} ,
                               },
                 },
every edge/.append style = {ma, postaction={decorate}, semithick}
                 ]
% from botom to top
% star coordinate
\coordinate             (a);
% nodes (ziga-zag
\node (a1)  [dot, label=below:$\alpha_1$,  right=of a];
\node (b1)  [dot, label=$\beta_{1}$,       above right=of a1]; 
\node (a2)  [dot, label=below:$\alpha_{2}$,below right=of b1];
\node (b2)  [dot, label=$\beta_{2}$,       above right=of a2]; 
\node (a3)  [dot, label=below:$\alpha_{3}$,below right=of b2];
\node (b3)  [dot, label=$\beta_{3}$,       above right=of a3];
% end coordinates
\coordinate (b) at (a |- b1);
% end coordinates
\coordinate[right=of b3,label=right:$\beta$]        (bb);
\coordinate[right=of b3 |- a3,label=right:$\alpha$] (aa); 
% continuation dashed lines
\draw[dashed] (a) -- (a1)   (a3) -- (aa)
              (b) -- (b1)   (b3) -- (bb);
% arrows
\draw[ma,dashed]        (a1) edge (b1)  (a3) edge (b3);
%
\draw   (b1) edge (a2)  (a2) edge (b2)  (b2) edge (a3) 
        (a3) edge (a2)  (a2) edge (a1) 
        (b3) edge (b2)  (b2) edge (b1);
    \end{tikzpicture}
\end{document}