Ordered tuples of ordered tuples

Question

The following source includes a definition of \opair for typesetting ordered pairs, or tuples, that use angle-brackets as the initial and terminal delimiters; that definition appears in https://tex.stackexchange.com/a/592421/13492. (I shrank the 3mu spacings in the original.)

\documentclass{article}
\usepackage{amsmath}
\ExplSyntaxOn
%
\NewDocumentCommand{\opair}{m}
 {
  \langle\mspace{1mu}
  \kaiserkatze_opair:n { #1 }
  \mspace{1mu}\rangle
 }
\seq_new:N \l__kaiserkatze_opair_items_seq
\tl_new:N \l__kaiserkatze_opair_first_tl
\cs_new_protected:Nn \kaiserkatze_opair:n
 {
  \seq_set_from_clist:Nn \l__kaiserkatze_opair_items_seq { #1 }
  % set off the first item and insert it
  \seq_pop_left:NN \l__kaiserkatze_opair_items_seq \l__kaiserkatze_opair_first_tl
  \tl_use:N \l__kaiserkatze_opair_first_tl
  % now insert commas and allowed breaks
  \seq_map_indexed_inline:Nn \l__kaiserkatze_opair_items_seq
   {
    % comma and thin space
    ,\mspace{1mu plus 1mu minus 1mu}
    % but \allowbreak only after the second item and before the penultimate
    \int_compare:nT { 1 < ##1 < \seq_count:N \l__kaiserkatze_opair_items_seq }
     { \allowbreak }
    % the item
    ##2
   }
 }
\ExplSyntaxOff
\begin{document}
$\opair{1,2}$,  $(1,2)$
\begin{gather}
\opair{\opair{1,2}, \opair{3,4}}
\
\bigl((1,2), (3,4)\bigr)
\end{gather}
\end{document}

Compare the output with ordinary use of parentheses and big parentheses for the same thing:

There are two limitations there:

1. In the case of an ordered pairs of ordered pairs, the parsing is contrary to the intended meaning.
2. As it stands, the definition of \opair does not allow for having larger angle brackets, as contrasted with using \bigl( ... \bigr) in the case of parentheses.

How can these limitations be overcome?

Note that for the second issue, it would not suffice to use \left\langle...\right\rangle. I want to be able to control the size of the angle brackets more finely, so as to use also \big\langle...\big\rangle and even \bigg\langle...\bigg\rangle. For this, it would be OK to have separate macros, named, say, \bigopair and \biggopair along with \leftrighopair.

Related: big angle brackets

Answer 1

The problem with \opair{\opair{1,2}, \opair{3,4}} is not due to parsing(*) but to recursion and assignments: when the second \opair is processed, it sets \l__kaiserkatze_opair_items_seq to a sequence containing the items 1 and 2, which disturbs what the outer \opair was doing: that one now sees an “incorrect” value of \l__kaiserkatze_opair_items_seq. Since the assignments done in \kaiserkatze_opair:n are all local, this can be fixed by wrapping the replacement text of \kaiserkatze_opair:n in \group_begin: ... \group_end:.

For the problem of delimiter sizes, you can make \opair accept an optional argument that gives the desired size. Here is one way using mathtools' \DeclarePairedDelimiterX:

\documentclass{article}
\usepackage{amsmath}
\usepackage{mathtools}
\ExplSyntaxOn
\DeclarePairedDelimiterX { \opair } [1] { \langle } { \rangle }
  {
    \mspace { 1mu }
    \kaiserkatze_opair:n {#1}
    \mspace { 1mu }
  }
\seq_new:N \l__kaiserkatze_opair_items_seq
\tl_new:N \l__kaiserkatze_opair_first_tl
\cs_new_protected:Nn \kaiserkatze_opair:n
 {
  \group_begin:
  \seq_set_from_clist:Nn \l__kaiserkatze_opair_items_seq {#1}
  % set off the first item and insert it
  \seq_pop_left:NN \l__kaiserkatze_opair_items_seq \l__kaiserkatze_opair_first_tl
  \tl_use:N \l__kaiserkatze_opair_first_tl
  % now insert commas and allowed breaks
  \seq_map_indexed_inline:Nn \l__kaiserkatze_opair_items_seq
   {
    % comma and thin space
    ,\mspace { 1mu plus 1mu minus 1mu }
    % but \allowbreak only after the second item and before the penultimate
    \int_compare:nT { 1 < ##1 < \seq_count:N \l__kaiserkatze_opair_items_seq }
     { \allowbreak }
    % the item
    ##2
   }
  \group_end:
 }
\ExplSyntaxOff
\begin{document}
$\opair{1,2}$,  $(1,2)$\quad
$\opair[\big]{1,2}$,  $\bigl(1,2\bigr)$
\begin{gather}
\opair{\opair{1,2}, \opair{3,4}} \
\opair[\big]{\opair{1,2}, \opair{3,4}} \
\opair[\Big]{\opair[\big]{1,2}, \opair[\big]{3,4}} \
% The starred form causes \left and \right to be used:
\opair{\opair{\frac{1}{2},2}, \opair{3,4}}
\end{gather*}
\end{document}

(*) Like macro arguments, items in a clist are always brace-balanced.