Nesting command definitions more than two levels

Question

How do you access macro arguments if the macros are nested by more than two levels?

To explain it in more detail: Creating a new command in another command (nesting command definitions), it is possible to implement something like partial application of a command. This is useful for example if a command, which is often called, has two arguments and one changes less frequently than the other:

\documentclass{minimal}
\begin{document}
    \newcommand{\one}[1]{%
        \newcommand{\two}[1]{#1, ##1}%
    }
    \one{a} % Partial application, nothing gets printed
    \two{b} \quad \two{c}
\end{document}

enter image description here

Unfortunately this does not work, if there is another level of nesting involved:

\documentclass{minimal}
\begin{document}
    \newcommand{\one}[1]{%
        \newcommand{\two}[1]{%
            \newcommand{\three}[1]{#1, ##1, ###1}%
        }
    }
    \one{a} % First partial application -> Error
    \two{b} % Second partial application
    \three{c}
\end{document}

The error occurs in the marked line with the following message:

[LaTeX] Line 8: Illegal parameter number in definition of \two.<to be read again>                    a
l.0 \one{c}
         % First partial application -> Error

As substituting \one{a} by \one{1} in the above code leads to the output 1, b, b, I think that ###1 is interpreted as ##{#1}. So ###1 is expanded to ##a or ##1, respectively. The first one obviously fails, where the second one compiles but leads to an unexpected result. I tested the commands from xparse, too, and the problem seems to be identical.

To rephrase the question above: How do you have ###1 interpreted as {###1} instead of ##{#1} when using the nested command definitions? Why is ##1 in the first example interpreted as {##1} instead of #{#1}?

I know that it is possible to avoid the third level of nesting by some sort of currying as in the following example.

\documentclass{minimal}
\begin{document}
    \newcommand{\one}[1]{%
        \newcommand{\two}[1]{\helper{#1, ##1}}%
    }
    \newcommand{\helper}[1]{%
        \newcommand{\three}[1]{%#1, ##1}%
    }
    \one{1} % First partial application
    \two{b} % Second partial application by currying
    \three{c}
\end{document}

enter image description here

It seems less intuitive and more complicated to me to use two levels of nesting twice instead of three levels of nesting once and I would like to understand how the third level of nesting is used.

You need four #s in the definition of \three: something like \newcommand{\three}[1]{#1, ##1, ####1}. — Gonzalo Medina, Apr 13 '12 at 18:53
Thanks a lot, that works indeed. Can you please add an answer, so that I can vote for it? I am also very interested in an explanation why that works … — Patrick Häcker, Apr 13 '12 at 18:57
Related, which also gives a similar explanation as in Gonzalo's answer: What is the meaning of double pound symbol (##1) in an argument? — Werner, Apr 13 '12 at 19:13
Thanks, I found a lot of related questions, but I missed this one. — Patrick Häcker, Apr 13 '12 at 19:22

Gonzalo Medina · Accepted Answer · 2012-04-13T19:09:14.117

Use four #s in the second nesting level:

\documentclass{article}
\begin{document}
    \newcommand{\one}[1]{%
        \newcommand{\two}[1]{%
            \newcommand{\three}[1]{#1, ##1, ####1}%
        }
    }
    \one{a} 
    \two{b} 
    \three{c}
\end{document}

enter image description here

The following quote from TeX by Topic gives an explanation (further details in section 11.5.5 The macro parameter character):

When TEX’s input processor scans a macro definition text, it inserts a parameter token for any occurrence of a macro parameter character followed by a digit. In effect, a parameter token in the replacement text states ‘insert parameter number such and such here’. Two parameter characters in a row are replaced by a single one.

Nesting command definitions more than two levels

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