Including braces and labels in Young tableaux

Question

Today I decided that I needed some practice in order to become more familiar with Young tableaux. I have the following piece of code -which returns the diagrams that I wanted-

\documentclass[11pt]{article}
%\usepackage{showkeys}
\usepackage{epsfig,amssymb,amsmath,psfrag,hyperref}
\usepackage{mathtools}
\usepackage[dvipsnames]{xcolor}
\usepackage[vcentermath]{youngtab} %Young boxes
\usepackage{ytableau} %Young diagrams
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
Let's do a fun exercise with Young tableaux 

\ytableausetup
{mathmode, boxsize=1.75em}
\begin{align}
    \begin{aligned}
\begin{ytableau}
 \\
\end{ytableau}
&&&
%%%%%%%%%%%%
\begin{ytableau}
1 &    \\
2   \\
\none[\vdots] \\
\scriptstyle N-2 \\
\scriptstyle N-1
\end{ytableau}
&&&
%%%%%%%%%%%%
&\begin{ytableau}
1 & & \none[\cdots] & & n \\
\end{ytableau} 
&&&
%%%%%%%%%%%%
&\begin{ytableau}
1  \\
2  \\
\none[\vdots] \\
 \\
n
\end{ytableau} 
\\[30pt]
%%%%%%%%%%%%
\begin{ytableau}
 &    \\
 &  \\
\end{ytableau}
&&&
%%%%%%%%%%%%
\begin{ytableau}
 &    \\
   \\
\end{ytableau}
&&&
%%%%%%%%%%%%
&\begin{ytableau}
1 &   \\
2 \\
\none[\vdots] \\
\scriptstyle N-3 \\
\scriptstyle N-2
\end{ytableau}
&&&
%%%%%%%%%%%%
\begin{ytableau}
1 &   \\
2 &   \\
\none[\vdots] \\
\scriptstyle N-3 \\
\scriptstyle N-2
\end{ytableau}
    \end{aligned} 
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

What I would like to understand is how to create the above but in the following way:

I tried to do this by following some answers and suggestions that I found here, but I failed spectacularly. This is what I tried

\ytableausetup
{mathmode, boxsize=1.75em}
\begin{align}
    \begin{aligned}
\begin{ytableau}
 \\
\end{ytableau}
&&&
%%%%%%%%%%%%
\left\{
\begin{ytableau}
1 &    \\
2   \\
\none[\vdots] \\
\scriptstyle N-2 \\
\scriptstyle N-1
\end{ytableau}
\right.
&&&
%%%%%%%%%%%%
&\begin{ytableau}
1 & & \none[\cdots] & & n \\
\end{ytableau} 
&&&
%%%%%%%%%%%%
&\begin{ytableau}
1  \\
2  \\
\none[\vdots] \\
 \\
n
\end{ytableau} 
\\[30pt]
%%%%%%%%%%%%
\begin{ytableau}
 &    \\
 &  \\
\end{ytableau}
&&&
%%%%%%%%%%%%
\begin{ytableau}
 &    \\
   \\
\end{ytableau}
&&&
%%%%%%%%%%%%
&\begin{ytableau}
1 &   \\
2 \\
\none[\vdots] \\
\scriptstyle N-3 \\
\scriptstyle N-2
\end{ytableau}
&&&
%%%%%%%%%%%%
\begin{ytableau}
1 &   \\
2 &   \\
\none[\vdots] \\
\scriptstyle N-3 \\
\scriptstyle N-2
\end{ytableau}
    \end{aligned} 
\end{align}

Any ideas?

Answer 1

Here is a possible way with TikZ. To make life easier, this answer comes with a style Young tableau and some additional keys like vdots and cdots for the dotted cells. The sizes of the cells and the dots are stored in pgf keys.

\documentclass[11pt]{article}
\usepackage{mathtools}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.pathreplacing,calligraphy,matrix}
\usepackage{hyperref}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tikzset{Dotted/.style={% https://tex.stackexchange.com/a/52856/194703
    line width=\pgfkeysvalueof{/tikz/Young/dot size},
    dash pattern=on 0.001\pgflinewidth off #1,line cap=round,
    shorten <=#1},Dotted/.default=3pt,
    vdots/.style={draw=none,path picture={
     \draw let \p1=(path picture bounding box.north),
        \p2=(path picture bounding box.south) in
        [Dotted={(\y1-\y2)/4}]
        (\p1) -- (\p2);
    }},
    cdots/.style={draw=none,path picture={
     \draw let \p1=(path picture bounding box.east),
        \p2=(path picture bounding box.west) in
        [Dotted={(\x1-\x2)/4}]
        (\p2) -- (\p1);
    }},
    Young tableau/.style={matrix of math nodes,nodes in empty cells,
    nodes={draw,minimum size=\pgfkeysvalueof{/tikz/Young/cell size},inner sep=0.5pt},
    column sep=-\pgflinewidth,row sep=-\pgflinewidth},
    Young/.cd,cell size/.initial=1.75em,
    dot size/.initial=1.2pt
    }
\begin{document}
\[\vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt1){ \\};
 \end{tikzpicture}}}
 \qquad
 \vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt2){ & \\
   \\
   \\
   |[vdots]| \\
   \\};
  \draw[thick,decoration={calligraphic brace, amplitude=6pt},decorate] 
  (yt2-5-1.south west) -- (yt2-2-1.north west)
  node[midway,above=1ex,sloped]{$N-2$};
 \end{tikzpicture}}}
 \qquad
 \vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt3){ & & |[cdots]| & & \\};
  \draw[thick,decoration={calligraphic brace, amplitude=6pt},decorate] 
  (yt3-1-5.south east) -- (yt3-1-1.south west)
  node[midway,below=1ex]{$n$};
 \end{tikzpicture}}}
 \qquad
 \vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt4){ \\ 
   \\ 
   |[vdots]| \\
   \\
   \\};
  \draw[thick,decoration={calligraphic brace, amplitude=6pt},decorate] 
  (yt4-1-1.north east) -- (yt4-5-1.south east)
  node[midway,right=1ex]{$n$};
 \end{tikzpicture}}}
\] 
\[\vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt5){ & \\ & \\ };
 \end{tikzpicture}}}
 \qquad
 \vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt6){ & \\  \\ };
 \end{tikzpicture}}}
 \qquad
 \vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt7){ & \\
   \\
   \\
   |[vdots]| \\
   \\};
  \draw[thick,decoration={calligraphic brace, amplitude=6pt},decorate] 
  (yt7-5-1.south west) -- (yt7-2-1.north west)
  node[midway,above=1ex,sloped]{$N-3$};
 \end{tikzpicture}}}
 \qquad
 \vcenter{\hbox{\begin{tikzpicture}
   \node[Young tableau](yt8){ & \\
    & \\
   \\
   \\
   |[vdots]| \\
   \\};
  \draw[thick,decoration={calligraphic brace, amplitude=6pt},decorate] 
  (yt8-6-1.south west) -- (yt8-3-1.north west)
  node[midway,above=1ex,sloped]{$N-4$};
 \end{tikzpicture}}}
\] 
\end{document}