General way to split a line by a rule of thumb and displaying results

Question

The following is to construct illustrations for concepts of frequency of a properly sampled outcome if any such frequency exists. Doing such illustrations is tedious and liable to error by hand ... so it's best to make a pair of macros and make a macro out of them that does the following:

Suppose I have a block of text with, say, 2 or more arbitrary letters or numbers,

axaaxxaaaxxaaaxxaaaaxxaxa

How does one make a macro that splits a line of text according to the following rule:

If the parameter to the macro is 2, take every second letter in each pair and juxtapose them. That is line 2. Delete these letters from the original line and the remainder is line 1. Display all 2 lines.

If the parameter to the macro is 3, take every second letter in each triple and juxtapose them. This is line 2. Take every third letter in each triple and juxtapose them. That is line 3. Delete these letters from the original line and the remainder is line 1. Display all 3 lines.

If I write

\brk{+}{1}{axaaxxaaaxxaaaxxaaaaxxaxa}

the equivalent manual code is

\begin{equation}
    \textrm{axaaxxaaaxxaaaxxaaaaxxaxa}
\end{equation}

If I write

\brk{+}{2}{axaaxxaaaxxaaaxxaaaaxxaxa}

the equivalent manual code is

\begin{equation}
    \textrm{aaxaaxaxaaxaa}
\end{equation}
\begin{equation}
    \textrm{xaxaxaaxaaxx}
\end{equation}

If I write

\brk{+}{3}{axaaxxaaaxxaaaxxaaaaxxaxa}

the equivalent manual code is

\begin{equation}
    \textrm{aaaxaxaxa}
\end{equation}
\begin{equation}
    \textrm{aaaxaxaxa}
\end{equation}
\begin{equation}
    \textrm{axxaxaaaa}
\end{equation}

And so on.

Now the - parameter chooses the second version of the macro and reverse this behavior.

If I write

\brk{-}{1}{axaaxxaaaxxaaaxxaaaaxxaxa}

the equivalent manual code is

\begin{equation}
    \textrm{axaaxxaaaxxaaaxxaaaaxxaxa}
\end{equation}

If I write

\brk{-}{2}{aaxaaxaxaaxaa}{xaxaxaaxaaxx}

the equivalent manual code is

\begin{equation}
    \textrm{axaaxxaaaxxaaaxxaaaaxxaxa}
\end{equation}

If I write

\brk{-}{3}{aaaxaxaxa}{aaaxaxaxa}{axxaxaaaa}

the equivalent manual code is

\begin{equation}
    \textrm{axaaxxaaaxxaaaxxaaaaxxaxa}
\end{equation}

And so on.

Yes, the ability to format the results into a specific environment for instance the equation environment or change the font or vertical spacing of the n lines in \{brk{+/-}{n}{axaaxxaaaxxaaaxxaaaaxxaxa is important as this is for figure purposes. Hopefully there is a way to make it not a problem for the macro.

BTW the result is this: if the strings are long enough and tending to a frequency of, say, x is 1/3 (iff the original string was randomly sampled as it must be for most scientific purposes) then in each of the two or three or more string made from it by any rule of thumb, the frequency of x is also tending to be 1/3 rather quickly. And if we sampled three parts of the same population each tending to a frequency of x being 1/3 and then added the strings together according to any systematic rule, the frequency of x tends again to 1/3.

Answer 1

If I understand your wish, you need to use the following macros:

\newcount\tmpnum
\def\sedef#1{\expandafter\edef\csname#1\endcsname}
\def\expandcsname#1{\expandafter\ifx\csname#1\endcsname\relax \else \csname#1\endcsname\fi}

\def\brk#1#2#3{\bgroup\def\brkLines{#2}\ifx+#1\tmpnum=0 \brkA#3{}%
   \else\tmpnum=1 \def\brkF{}\def\tmp{\brkC{#3}}\expandafter\tmp\fi
}
\def\brkA#1{\ifx$#1$\expandafter\brkB\else 
   \advance\tmpnum by1   
   \ifnum\tmpnum>\brkLines \tmpnum=1 \fi
   \sedef{brk:\the\tmpnum}{\expandcsname{brk:\the\tmpnum}#1}%
   \expandafter\brkA \fi
}
\def\brkB{\tmpnum=0
   \loop
      \advance\tmpnum by1
      $$\hbox{\csname brk:\the\tmpnum\endcsname}$$
      \ifnum\tmpnum<\brkLines\relax \repeat
   \egroup
}
\def\brkC #1{%
   \sedef{brk:\the\tmpnum}{#1}%
   \ifnum\tmpnum<\brkLines\relax \advance\tmpnum by1 \expandafter\brkC
   \else \expandafter\brkD \fi
}
\def\brkD{\tmpnum=0
   \loop
      \advance\tmpnum by1
      \ifnum\tmpnum>\brkLines \tmpnum=1 \fi
      \expandafter\ifx\csname brk:\the\tmpnum\endcsname \empty \tmpnum=0
      \else
         \edef\tmp{\csname brk:\the\tmpnum\endcsname}%
         \expandafter \brkE \tmp \brkF
      \fi 
      \ifnum\tmpnum>0 \repeat
   $$\hbox{\brkF}$$
   \egroup
}
\def\brkE#1#2\brkF{\edef\brkF{\brkF#1}\sedef{brk:\the\tmpnum}{#2}}


\brk{+}{3}{123abcuvw} % --> 1au // 2bv // 3cw

\brk{-}{3}{123}{abc}{uvw}  % --> 1au2bc3cw

\brk{+}{3}{axaaxxaaaxxaaaxxaaaaxxaxa}

\brk{-}{3}{aaaxaxaxa}{aaaxaxaxa}{axxaxaaaa}

A short summary, how it works. The \brk macro executes \brkA if the parameter is + (break to more lines) and executes \brkC if the parameter is - (merge to one line).

The \brkA reads token per token from input stream in the loop and appends it to macros \brk:num where num is 1,2,3,...,\brkLines, 1,2,3,...,\brkLines,...etc. The \brkA ends by executing \brkB: it prints contents of \brk:num macros from one to \brkLines in the display lines.

The inverse processing is done by \brkC which stores the n parameters (n=\brkLines) to the macros \brk:num and executes \brkD. This macro executes the loop where it reads and removes first token from \brk:num, where num is 1,2,3,...,\brkLines,1,2,3,...,\brkLines,etc. until first such macro is emty. The read tokens are accumulated to \brkF by help of \brkE. Finally the \brkF is printed in the display line.

Answer 2

Here's an application of expl3: for splitting I cycle the requested number of times through the input string, picking up the items corresponding to the cycle and forming a sequence; for rebuilding it's just the reverse: I cycle over the sequence elements picking up the first element, the second one and so on from each item.

\documentclass{article}
\usepackage{amsmath}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\breakstring}{mm}
 {
  \guido_breakstring:nn {#1} {#2}
 }
\NewDocumentCommand{\rebuildstring}{m}
 {
  \guido_rebuildstring:n { #1 }
 }

\int_new:N \l_guido_columns_int
\seq_new:N \l_guido_strings_seq
\tl_new:N \l_guido_temp_tl

\cs_new_protected:Npn \guido_breakstring:nn #1 #2
 {
  \seq_clear:N \l_guido_strings_seq
  \int_step_inline:nnnn { 1 } { 1 } { #1 }
   {
    \int_zero:N \l_guido_columns_int
    \tl_clear:N \l_guido_temp_tl
    \tl_map_inline:nn { #2 }
     {
      \int_incr:N \l_guido_columns_int
      \int_compare:nT { \l_guido_columns_int == ##1 }
       {
        \tl_put_right:Nn \l_guido_temp_tl { \mathrm{####1} }
       }
      \int_compare:nT { \l_guido_columns_int == #1 }
       { \int_zero:N \l_guido_columns_int }
     }
    \seq_put_right:NV \l_guido_strings_seq \l_guido_temp_tl
   }
  \begin{gather*}
  \seq_use:Nn \l_guido_strings_seq { \\ }
  \end{gather*}
 }

\cs_new_protected:Npn \guido_rebuildstring:n #1
 {
  \seq_set_split:Nnn \l_guido_strings_seq { ; } { #1 }
  \tl_set:Nx \l_guido_temp_tl { \seq_item:Nn \l_guido_strings_seq { 1 } }
  \int_set:Nn \l_guido_columns_int { \tl_count:N \l_guido_temp_tl }
  \tl_clear:N \l_guido_temp_tl
  \int_step_inline:nnnn { 1 } { 1 } { \l_guido_columns_int }
   {
    \seq_map_inline:Nn \l_guido_strings_seq
     {
      \tl_put_right:Nx \l_guido_temp_tl { \exp_not:N \mathrm{ \tl_item:nn { ####1 } { ##1 } } }
     }
   }
  \begin{equation*}
  \tl_use:N \l_guido_temp_tl { \\ }
  \end{equation*}
 }

\ExplSyntaxOff

\begin{document}
\breakstring{3}{12312312312}

\rebuildstring{1111;2222;333}

\section{Split}

One:
\breakstring{1}{axaaxxaaaxxaaaxxaaaaxxaxa}
Two:
\breakstring{2}{axaaxxaaaxxaaaxxaaaaxxaxa}
Three:
\breakstring{3}{axaaxxaaaxxaaaxxaaaaxxaxa}

\section{Rebuild}

One:
\rebuildstring{axaaxxaaaxxaaaxxaaaaxxaxa}
Two:
\rebuildstring{aaxaaxaxaaxaa;xaxaxaaxaaxx}
Three:
\rebuildstring{aaaxaxaxa;xxaxaaaa;axaaxaxx}

\end{document}