Different alignments in {align*} formula

Question

I have problem with creating formula with different alignments. I want to create some kind of 'permutation dependings diagram'. Code witch I created so far:

\begin{align*} 
    &(1, 2, 3, 4, 5, 6) \\
    &(2, 5, 1, 6, 4, 3) \xleftrightarrow{\text{s}} (3, 1, 6, 5, 2, 4)\\
    \updownarrow{\text{o}}
\end{align*}

Output:

Expected output:

I want to have permutations in two vertical lines - all "(...)" have to be in two vertical lines. Second thing is that I don't know how can I make vertical arrow in the middle of permutations - in the middle of text: "(2, 5, 1, 6, 4, 3)" and second one "(3, 1, 6, 5, 2, 4)"

There is any simple way to prepare diagram like this on handwritten picture?

Answer 1

There does not seem to be a need for align here, as you can set this using an array and choose your alignments:

\documentclass{article}

\usepackage{amsmath,extarrows}

\begin{document}

\[
  \renewcommand{\arraystretch}{1.2}% Spread array out vertically
  \begin{array}{ *{3}{c} }
    (1, 2, 3, 4, 5, 6) \\
    (2, 5, 1, 6, 4, 3) & \xleftrightarrow{\text{s}} & (3, 1, 6, 5, 2, 4) \\
    \updownarrow \raisebox{.1em}{\scriptsize o} & & \updownarrow \raisebox{.1em}{\scriptsize o} \\
    (5, 3, 6, 2, 4, 1) &                            & (5, 1, 4, 6, 3, 2) \\
    \updownarrow \raisebox{.1em}{\scriptsize s} & & \updownarrow \raisebox{.1em}{\scriptsize s} \\
    (6, 4, 2, 5, 1, 3) &                            & (2, 6, 5, 3, 1, 4) \\
          \vdots       &                            &       \vdots       \\
    (6, 5, 4, 3, 2, 1) & \xleftrightarrow{\text{s}} & (6, 5, 4, 3, 2, 1)
  \end{array}
\]

\end{document}

You can play around with the spacing between columns or rows.

Answer 2

If it can rest on the same page, array seems to be the simplest solution (a starting point}:

\documentclass{article}
\usepackage{amsmath}
\usepackage{extarrows} 

\begin{document}

\[
\begin{array}{ccc}
    (1, 2, 3, 4, 5, 6)&& \\
    (2, 5, 1, 6, 4, 3)& \xleftrightarrow{\text{s}}& (3, 1, 6, 5, 2, 4)\\
    \updownarrow{\text{o}}&&\\
    \text{etc.} & \text{etc.}&\text{etc.}\\
    \vdots&&\vdots
\end{array}
\]

\end{document}