Staircase paths through young tableaux

Question

I would like to draw a staircase path in a young tableau that uses the sides of the boxes in the tableau as steps.

For example, in the tableau

\documentclass[border=2mm]{standalone}
\usepackage{ytableau}
\begin{document}
\ytableausetup
{boxsize=1.25em}
\ytableausetup
{aligntableaux=top}
\begin{ytableau}
*(blue!40) 1& *(blue!40) 3 &*(blue!40) 5 &7&9\\
*(blue!40) 2 & *(blue!40) 4 &8&10&11
\end{ytableau}
\end{document}  

I would like to have the bottom side of the box with the entry 2 as the first step, then bottom of box 4, then right side of box 4, bottom of box 5, right side of box 5, top box 7, top box 9.

The colors are not important, they're there for convenience, as long as there is this path drawn across the tableaux.

Any ideas?

Answer 1

tikz package can be used for this purpose. We can consider this tableau as the text within a node and draw the staircase consequently. We use varwidth package here to made the node as compact as possible. The proposed code is as follows:

\documentclass[border=2mm]{standalone}
\usepackage{ytableau,tikz,varwidth}
\usetikzlibrary{calc}
\begin{document}

\begin{tikzpicture}[inner sep=0in,outer sep=0in]
\node (n) {\begin{varwidth}{5cm}{
\begin{ytableau}
*(blue!40) 1 & *(blue!40) 3 & *(blue!40) 5 &7  &9\\
*(blue!40) 2 & *(blue!40) 4 & 8            &10 &11
\end{ytableau}}\end{varwidth}};
\draw[thick,orange] (n.south west)--([xshift=1.09cm]n.south west)--([xshift=1.09cm]n.west)--([xshift=-1.09cm]n.east)--([xshift=-1.09cm]n.north east)--(n.north east);
\end{tikzpicture}
\end{document}

With the expected output:

As per Combin's comment, we can perform this solution to an arbitrary tableau. It is only a matter of knowing the cell size. Here is another example of a skew tableau:

\documentclass[border=2mm]{standalone}
\usepackage{ytableau,tikz,varwidth}
\usetikzlibrary{calc}
\begin{document}

\begin{tikzpicture}[inner sep=0in,outer sep=0in]
\node (n) {\begin{varwidth}{5cm}{
\ytableausetup{notabloids}
\begin{ytableau}
\none & \none & *(blue!40)1 & 2 \\
\none & *(blue!40)1 & 2 \\
*(blue!40)1 & 2 \\
2
\end{ytableau}}\end{varwidth}};
\draw[very thick,orange] ([yshift=-.5*1.09cm]n.west)--++(.5*1.09,0)--++(0,.5*1.09)--++(.5*1.09,0)--++(0,.5*1.09)--++(.5*1.09,0)--++(0,.5*1.09)--++(.5*1.09,0);
\end{tikzpicture}
\end{document}

Now the output is like this:

As a variant, try the following tikz solution. It requires two runs one to calculate the positions and a second to draw. Here is the proposed code (inspired by an answer by Peter Grill to this question):

\documentclass[border=2mm]{standalone}
\usepackage{ytableau}
\usepackage{tikz}
\begin{document}

\newcommand{\tikzmark}[2]{\tikz[overlay,remember picture,baseline] \node [anchor=base] (#1) {$#2$};}

\newcommand{\DrawVLine}[3][]{%
  \begin{tikzpicture}[overlay,remember picture]
    \draw[shorten >=0.2mm,shorten <=-0.1mm, #1] ([xshift=0.1mm]#2.south east) -- ([xshift=0.1mm]#3.north east);
  \end{tikzpicture}
}

\newcommand{\DrawHLine}[3][]{%
  \begin{tikzpicture}[overlay,remember picture]
    \draw[shorten <=-0.2mm,shorten >=-0.1mm, #1] (#2.south west) -- (#3.south east);
  \end{tikzpicture}
}

\newcommand{\DrawHLineAbove}[3][]{%
  \begin{tikzpicture}[overlay,remember picture]
    \draw[shorten >=-0.2mm, #1] ([yshift=-0.3mm]#2.north east) -- ([yshift=-0.31mm]#3.north east);
  \end{tikzpicture}
}

\ytableausetup
{boxsize=1.25em}
\ytableausetup
{aligntableaux=top}
\begin{ytableau}
 1&  3 &\tikzmark{b-5}{5}&7&\tikzmark{b-9}{9}\\
\tikzmark{b-2}{2} & \tikzmark{b-4}{4}&8&10&11
\end{ytableau}

\DrawHLine[blue, thick]{b-2}{b-4}
\DrawVLine[blue, thick]{b-4}{b-4}
\DrawHLine[blue, thick]{b-5}{b-5}
\DrawVLine[blue, thick]{b-5}{b-5}
\DrawHLineAbove[blue, thick]{b-5}{b-9}

\end{document} 

The output to this is:

Answer 2

[Edit: improved syntax and illustrating how to to cope with different shapes, answering the OP's query in the comments..]

I agree with AboAmmar that tikz is a good approach. Here is a slightly shorter solution where the main difference is that it uses tikz to draw the tableau, resulting in a number of shortcuts. Here is the result of, first, drawing a tableau normally and, secondly, inserting this same code into a another tikzpicture environment so that the blue lines can be drawn:

I have rewritten my original code so that it looks nicer and is easier to use. I have also added the third example above to show how to do a three-rowed tableau -- arbitrary shapes are fine.

The example code below defines a Tableau environment for drawing a tableau and that then allows arbitrary tikz code to appear after the tableau. There is also a "shortcut" macro \tableau that draws an ordinary tableau without any additional decorations.

With both the Tableau environment and \tableau macro the entries of the tableau are specified as "comma separated lists" like `{{1,3,5,7,9},{2,4,8,10,11}}. This defines a two-rowed tableau with 1 3 5 7 9 in the first row and 2,4,8,10,11 in the second row. You can specify an arbitrary number of rows and columns with any reasonable entries this way.

Here is the full latex code:

\documentclass{article}
\usepackage{tikz}
\usepackage{xparse}

\newcount\tableauRow
\newcount\tableauCol
\def\tableauDim{0.4}
\newenvironment{Tableau}[1]{%
  \tikzpicture[scale=0.4,draw/.append style={thick,black},
                      baseline=(current bounding box.center)]
    % now draw the tableau
    \tableauRow=-1.5
    \foreach \Row in {#1} {
       \tableauCol=0.5
       \foreach\k in \Row {
         \draw[thin](\the\tableauCol,\the\tableauRow)rectangle++(1,1);
         \draw[thin](\the\tableauCol,\the\tableauRow)+(0.5,0.5)node{$\k$};
         \global\advance\tableauCol by 1
       }
       \global\advance\tableauRow by -1
    }
}{\endtikzpicture}
\newcommand\tableau[1]{\begin{Tableau}{#1}\end{Tableau}}

\begin{document}

\begin{tabular}{l@{\qquad}l@{\qquad}l}
  \tableau{{1,3,5,7,9},{2,4,8,10,11}}
&
  \begin{Tableau}{{1,3,5,7,9},{2,4,8,10,11}}
      \draw[very thick,blue] (0,-2)--(2,-2)--(2,-1)--(3,-1)--(3,0)--(5,0);
  \end{Tableau}
&
  \begin{Tableau}{{1,5,7,9},{2,4,10,11},{3,8,14}}
    \draw[very thick,blue](0,-3)--(2,-3)--(2,-2)--(3,-2)--(3,-1)--(4,-1)--(4,0);
  \end{Tableau}
\end{tabular}

\end{document}

Btw, I have changed the code so that the boxes not sit on the "obvious" integer lattice points.

It's also easy to colour the boxes in this way and do many other things. I use these macros in my own papers -- although the trick of using an environment for decorating the tableaux is something that I only just thought of when editing this, so thanks:)