Horizontal alignment of row vectors inside flalign and cases environments

Question

I wrote this code

\begin{flalign*}
    t\hphantom{^{\prime}}&=
        \begin{cases}
            \begin{pmatrix} 1 & 2 & ... & \frac{p}{2} & \frac{p}{2}+1 & ... & p \end{pmatrix} & \text{if $p$ even,} \\
            \begin{pmatrix} 1 & 2 & ... & \frac{p-1}{2} & \frac{p-1}{2}+1 & ... & p \end{pmatrix} & \text{if $p$ odd,}
        \end{cases}&&\\ 
    t^{\prime}&=
        \begin{cases}
            \begin{pmatrix} \frac{p}{2}+1 & \frac{p}{2}+2 & ... & p & 1 & ... & \frac{p}{2} \end{pmatrix} & \text{if $p$ even,} \\
            \begin{pmatrix} \frac{p-1}{2}+1 & \frac{p-1}{2}+2 & ... & p & 1 & ... & \frac{p-1}{2} \end{pmatrix} & \text{if $p$ odd.}
        \end{cases}&&
\end{flalign*}

which produces the following equations

How can I get respective elements of all row vectors to align horizontally?

Answer 1

You can use nicematrix:

\documentclass{article}
\usepackage{amsmath}
\usepackage{nicematrix}
\begin{document}
[
\begin{NiceArray}{@{} l @{\quad} l @{\qquad} l @{}}
  \Block{2-1}{t\hphantom{'} = {}} &
  \bigl(\begin{matrix}
    1 & 2 & \dots & \frac{p}{2} & \frac{p}{2}+1 & \dots & p
  \end{matrix}\bigr) &
  \text{$p$ even,}
  \[1ex]
  &
  \bigl(\begin{matrix}
    1 & 2 & \dots & \frac{p-1}{2} & \frac{p-1}{2}+1 & \dots & p
  \end{matrix}\bigr) & \text{$p$ odd,}
  \ \
  \Block{2-1}{t' = {}} &
  \bigl(\begin{matrix}
    \frac{p}{2}+1 & \frac{p}{2}+2 & \dots & p & 1 & \dots & \frac{p}{2}
  \end{matrix}\bigr) & \text{$p$ even,}
  \[1ex]
  &
  \bigl(\begin{matrix}
    \frac{p-1}{2}+1 & \frac{p-1}{2}+2 & \dots & p & 1 & \dots & \frac{p-1}{2}
  \end{matrix}\bigr) &
  \text{$p$ odd.}
\CodeAfter\SubMatrix{{1-2}{2-2}.\SubMatrix{{4-2}{5-2}.
\end{NiceArray}
]
\end{document}

If you really want that the row vector entries are aligned,

\documentclass{article}
\usepackage{amsmath}
\usepackage{nicematrix}
\begin{document}
[
\begin{NiceArray}{
  @{}
  l
  @{\quad}
  r
  @{}
  ccccccc
  @{}
  l
  @{\qquad}
  l
  @{}
}
  \Block{2-1}{t\hphantom{'} = {}} &
  \big( & 1 & 2 & \dots & \frac{p}{2} & \frac{p}{2}+1 & \dots & p & \big) &
  \text{$p$ even,}
  \[1ex]
  &
  \big( & 1 & 2 & \dots & \frac{p-1}{2} & \frac{p-1}{2}+1 & \dots & p & \big) &
  \text{$p$ odd,}
  \ \
  \Block{2-1}{t' = {}} &
  \big( & \frac{p}{2}+1 & \frac{p}{2}+2 & \dots & p & 1 & \dots & \frac{p}{2} & \big) &
  \text{$p$ even,}
  \[1ex]
  &
  \big( & \frac{p-1}{2}+1 & \frac{p-1}{2}+2 & \dots & p & 1 & \dots & \frac{p-1}{2} & \big) &
  \text{$p$ odd.}
\CodeAfter\SubMatrix{{1-2}{2-2}.\SubMatrix{{4-2}{5-2}.
\end{NiceArray}
]
\end{document}

Answer 2

The following solution sets each column element inside an \eqmathbox[<tag>], using the same <tag> for each column. eqparbox then figures out which element with the same <tag> is the widest and sets them all in a box of this maximum-width. This allows things to line up horizontally.

\documentclass{article}
\usepackage{amsmath,eqparbox}
% https://tex.stackexchange.com/a/34412/5764
\makeatletter
\NewDocumentCommand{\eqmathbox}{o O{c} m}{%
  \IfValueTF{#1}
    {\def\eqmathbox@##1##2{\eqmakebox[#1][#2]{$##1##2$}}}
    {\def\eqmathbox@##1##2{\eqmakebox{$##1##2$}}}
  \mathpalette\eqmathbox@{#3}
}
\makeatother
\begin{document}
Original nonalignment:
\begin{flalign}
  t\hphantom{^{\prime}}&=
    \begin{cases}
      \begin{pmatrix} 1 & 2 & \cdots & \frac{p}{2} & \frac{p}{2} + 1 & \cdots & p \end{pmatrix} & \text{if $p$ even,} \
      \begin{pmatrix} 1 & 2 & \cdots & \frac{p - 1}{2} & \frac{p - 1}{2} + 1 & \cdots & p \end{pmatrix} & \text{if $p$ odd,}
    \end{cases}&&\ 
  t^{\prime}&=
    \begin{cases}
      \begin{pmatrix} \frac{p}{2} + 1 & \frac{p}{2} + 2 & \cdots & p & 1 & \cdots & \frac{p}{2} \end{pmatrix} & \text{if $p$ even,} \
      \begin{pmatrix} \frac{p-1}{2} + 1 & \frac{p - 1}{2} + 2 & \cdots & p & 1 & \cdots & \frac{p - 1}{2} \end{pmatrix} & \text{if $p$ odd.}
    \end{cases}&&
\end{flalign}
Updated horizontal alignment:
\begingroup
\setlength{\arraycolsep}{0.5\arraycolsep}%
\begin{flalign}
  t\hphantom{^{\prime}} &=
    \begin{cases}
      \begin{pmatrix} 
        \eqmathbox[c1]{1} & \eqmathbox[c2]{2} & \eqmathbox[c3]{\cdots} & \eqmathbox[c4]{\frac{p}{2}} & 
          \eqmathbox[c5]{\frac{p}{2} + 1} & \eqmathbox[c6]{\cdots} & \eqmathbox[c7]{p} 
      \end{pmatrix} & \text{if $p$ even,} \
      \begin{pmatrix}
        \eqmathbox[c1]{1} & \eqmathbox[c2]{2} & \eqmathbox[c3]{\cdots} & \eqmathbox[c4]{\frac{p - 1}{2}} & 
          \eqmathbox[c5]{\frac{p - 1}{2} + 1} & \eqmathbox[c6]{\cdots} & \eqmathbox[c7]{p}
      \end{pmatrix} & \text{if $p$ odd,}
    \end{cases}&& \ 
  t^{\prime} &=
    \begin{cases}
      \begin{pmatrix}
        \eqmathbox[c1]{\frac{p}{2} + 1} & \eqmathbox[c2]{\frac{p}{2} + 2} & \eqmathbox[c3]{\cdots} & \eqmathbox[c4]{p} & 
          \eqmathbox[c5]{1} & \eqmathbox[c6]{\cdots} & \eqmathbox[c7]{\frac{p}{2}}
      \end{pmatrix} & \text{if $p$ even,} \
      \begin{pmatrix}
        \eqmathbox[c1]{\frac{p - 1}{2} + 1} & \eqmathbox[c2]{\frac{p -1}{2} + 2} & \eqmathbox[c3]{\cdots} & \eqmathbox[c4]{p} & 
          \eqmathbox[c5]{1} & \eqmathbox[c6]{\cdots} & \eqmathbox[c7]{\frac{p - 1}{2}}
      \end{pmatrix} & \text{if $p$ odd.}
    \end{cases}&&
\end{flalign}
\endgroup
\end{document}

An alternative would be to set the elements in a fixed-width column array. It's just convenient to let TeX figure out the widths of each column.