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\usepackage{enumerate}
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\usepackage[all]{xy}
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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 31 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

%$\;$

\noindent\textbf{Topics}: Catalan numbers


\begin{enumerate}


\item Everyone in class will be assigned a unique Catalan object from
  the following list.
  \[
  \hspace*{-13mm}
\begin{tikzpicture}[scale=1,every node/.style={font=\small}]
    \tikzstyle{e} = [bend left=45] 
    \def\x{4.8} 
    \def\y{4.5} 
    %%
    \begin{scope}[shift={(0,.15)},scale=.2]
    \draw (0,0) circle (1cm);
    \draw (2,0) circle (1cm);
    \draw (4,0) circle (1cm);
    \draw (6,0) circle (1cm);
    %%
    \draw (1,1.732) circle (1cm);
    \draw (3,1.732) circle (1cm);
    \draw (5,1.732) circle (1cm);
    %%
    \draw (2,3.464) circle (1cm);
    \draw (4,3.464) circle (1cm);
    %%
    \draw (3,5.196) circle (1cm);
    \node at (3,-2.5) {coin stacks;};
    \node at (3,-4.5) {$n$ in bottom row};
    \end{scope}
    %%
    \begin{scope}[shift={(3,0)},scale=.2]
      \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, 
        minimum size=.65mm]
      \node[v] (v1) at (1,0) {};
      \node[v] (v2) at (2,0) {};
      \node[v] (v3) at (3,0) {};
      \node[v] (v4) at (4,0) {};
      \node[v] (v5) at (5,0) {};
      \node[v] (v6) at (6,0) {};
      \node[v] (v7) at (7,0) {};
      \node[v] (v8) at (8,0) {};
      \node[v] (v9) at (9,0) {};
      \node[v] (v10) at (10,0) {};
      \draw[e] (v1) to (v6); \draw[e] (v2) to (v7); \draw[e] (v3) to (v8); 
      \draw[e] (v4) to (v9); \draw[e] (v5) to (v10);
      \node at (5,-2) {non-nesting};
      \node at (5,-4) {matchings of $[2n]$};
    \end{scope}
    %%
    \begin{scope}[shift={(6.5,0)},scale=.2]
      \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, 
        minimum size=.65mm]
      \node[v] (v1) at (1,0) {};
      \node[v] (v2) at (2,0) {};
      \node[v] (v3) at (3,0) {};
      \node[v] (v4) at (4,0) {};
      \node[v] (v5) at (5,0) {};
      \node[v] (v6) at (6,0) {};
      \node[v] (v7) at (7,0) {};
      \node[v] (v8) at (8,0) {};
      \node[v] (v9) at (9,0) {};
      \node[v] (v10) at (10,0) {};
      \draw[e] (v1) to (v10); \draw[e] (v2) to (v9); \draw[e] (v3) to (v8); 
      \draw[e] (v4) to (v7); \draw[e] (v5) to (v6);
      \node at (5,-2) {non-crossing};
      \node at (5,-4) {matchings of $[2n]$};      
    \end{scope}
    %%
    \begin{scope}[shift={(10.5,0)},scale=.4,every label/.style={font=\scriptsize}]
      \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, 
        minimum size=1mm]
      \def\n{8} 
      \node[v,label=right:{$1$}] (1) at (0,0) {};
      \node[v,label=left:{$2$}] (2) at (-1,1) {};
      \node[v,label=right:{$3$}] (3) at (1,1) {};
      \node[v,label=left:{$4$}] (4) at (0,2) {};
      \node[v,label=right:{$5$}] (5) at (2,2) {};
      \node[v,label=left:{$6$}] (6) at (1,3) {};
      \node[v,label=right:{$7$}] (7) at (3,3) {};
      \node[v,label=left:{$8$}] (8) at (2,4) {};
      \draw (1)--(2)--(4)--(6)--(8)--(7)--(5)--(3)--(1);
      \draw (3)--(4); \draw (5)--(6);
      \node at (1,-1) {linear extensions};
      \node at (1,-1.9) {of $\bm{2\times n}$};
    \end{scope}
    %%
    \begin{scope}[shift={(14.5,.4)},scale=1]
      \ytableausetup{aligntableaux =top, nobaseline,boxsize=1.25em}
      \node at (0,0) {$\ytableaushort{1257,346}$};
      \node at (0,-.9) {SYT of shape};
      \node at (0,-1.3) {$(n,n)$ or $(n,n-1)$};
    \end{scope}
    %%---------------------------------------------------------
    \begin{scope}[shift={(1,-3.5)},scale=.4]
      \draw (0,0)--(1,0)--(1,1)--(2,1)--(2,2)--(3,2)--(3,3)--(4,3)--(4,4)--(0,4)--cycle;
      \draw (1,1)--(1,3)--(0,3);
      \draw (1,3)--(3,3);
      \draw (1,2)--(2,2);
      \node at (1,-.8) {tilings of $(n,\dots,n)$};
      \node at (1,-1.6) {with $n$ rectangles};      
    \end{scope}
    %%
    \begin{scope}[shift={(5.5,-2.7)},scale=.9]
        \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, 
          minimum size=.75mm]
        \def\n{8} 
        \def\radius{.75} 
        \def\x{5} 
        \def\y{4.5} 
        \foreach \i in {0,...,\n} {
        \node[v] (v\i) at ({360/\n * \i}:\radius) {};}
       \foreach \i in {1,...,\n} {
       \pgfmathtruncatemacro{\j}{\i-1}
        %\draw (v\i) to (v\j); 
        }
       \draw (v0) to (v1); \draw (v2) to (v3);
       \draw (v4) to (v7);
       \draw (v5) to (v6);
      \node at (0,-1.2) {$n$ non-intersecting};
      \node at (0,-1.6) {chords on $2n$ points};      
    \end{scope}
    %%
    \begin{scope}[shift={(8.25,-3.3)},scale=.5,every label/.style={font=\footnotesize}]
      \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, 
        minimum size=.85mm]
      \node[v] (v1) at (1,0) {};
      \node[v] (v2) at (2,0) {};
      \node[v] (v3) at (3,1) {};
      \node[v] (v4) at (4,1) {};
      \node[v] (v5) at (5,1) {};
      \node[v] (v6) at (6,0) {};
      \draw (v1)--(v2)--(v3)--(v4)--(v5)--(v6);
      \node at (3.5,-1) {peakless Motzkin paths;};
      \node at (3.5,-1.8) {$n$ up \& level steps};
    \end{scope}
    %%
    \begin{scope}[shift={(13.5,-3.3)},scale=.5,every label/.style={font=\footnotesize}]
      \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, 
        minimum size=.85mm]
      \draw (0,0)--(0,2.5);
      \draw (1,0)--(1,2.15);
      \draw (2,0)--(2,2.5);
      \draw (3,0)--(3,2.5);
      \draw (0,2.5)--(2,2.5);
      \node at (1.5,-1) {noncrossing Murasaki};
      \node at (1.5,-1.8) {diagrams; $n$ $\big|$'s};
    \end{scope}
    %%
\end{tikzpicture}
\]
For your particular object, carry out the following steps.
\begin{enumerate}
\item Use TikZ to make a visual of all $C_4=14$ examples for $n=4$. 
\item Make a list of ``natural'' combinatorial statistics, and compute
  their distributions for $n=4$.
\item Construct a bijection to one of the Catalan objects that we saw in class. 
  \end{enumerate}
  

\item Richard Stanley maintains a list of over $200$ combinatorial objects counted by the Catalan numbers. Over 160 of these are in his \emph{Catalan Addendum} (\url{https://math.mit.edu/~rstan/ec/catadd.pdf}). Pick two of these, different from your classmates. At least one should be ``visual'' (not just numeric sequences), and along with your Catalan object from Problem \#1, all three should seem distinctly different. 
  \begin{enumerate}
  \item Use TikZ to make a visual of all $C_4=14$ examples for $n=4$. 
  \item Give bijective proofs to show that these are all equivalent to
    each other. These should be self-contained; do not use other
    Catalan objects.
  \item Make a list of ``natural'' combinatorial statistics, and
    compute their distributions for $n=4$. What is the interpretation
    of these under your bijections?
  \end{enumerate}


 

\item Consider the bijection we saw in class from the triangulations of an $(n+2)$-gon, to the binary trees on $n$ vertices; an example is shown below.
  \[
\begin{tikzpicture}[scale=2]
 \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, minimum
   size=1.1mm]
    %%
   \begin{scope}[shift={(0,0)}]
   \def\n{12}        % number of sides
  \def\r{1}        % number of sides
  \def\shift{180/\n} % half-angle rotation = 15 degrees for n=12
  %%
  % define vertices
  \foreach \i in {0,...,11}{
    \coordinate (v\i) at ({\r*cos(360/\n*\i + \shift)},
                          {\r*sin(360/\n*\i + \shift)});
  }
  %%
\foreach \i in {0,...,11} {
    % compute next index mod n
    \pgfmathtruncatemacro{\j}{mod(\i+1,\n)}
    % conditional: if i==2 draw dashed, else solid
    \ifnum\i=2
      \draw[dashed] (v\i) to node[midway,above] {$e$} (v\j);
    \else
      \draw (v\i) -- (v\j);
    \fi
  }
  \draw (v0) -- (v2); \draw (v2) -- (v9); \draw (v9) -- (v0); \draw (v10) -- (v0); 
  \draw (v3) -- (v9); \draw (v3) -- (v7); \draw (v9) -- (v7); \draw (v4) -- (v6);
  \draw (v3) -- (v6);

  \node[v,label=above:{$v$}] (r) at (.1,.5) {};
  \node[v] (t1) at (-.27,0) {};
  \node[v] (t2) at (-.62,.1) {};
  \node[v] (t3) at (-.66,.5) {};
  \node[v] (t4) at (-.9,.23) {};
  \node[v] (t5) at (-.27,-.9) {};
  \node[v] (t6) at (.45,.1) {};
  \node[v] (t7) at (.65,-.45) {};
  \node[v] (t8) at (.9,-.23) {};
  \node[v] (t9) at (.65,.67) {};
  %%  
  \draw[dashed] (r)--(t1)--(t2)--(t3)--(t4);
  \draw[dashed] (t1)--(t5);
  \draw[dashed] (r)--(t6)--(t7)--(t8);
  \draw[dashed] (t6)--(t9);       
  \end{scope}
    %%
    \begin{scope}[shift={(2.75,.7)},scale=.35]
    \node[v] (r) at (0,0) {};
    \node[v] (t1) at (-1,-1) {};
    \node[v] (t2) at (-1.5,-2) {};
    \node[v] (t3) at (-2,-3) {};
    \node[v] (t4) at (-1.5,-4) {};
    \node[v] (t5) at (-.5,-2) {};
    \node[v] (t6) at (1,-1) {};
    \node[v] (t7) at (.5,-2) {};
    \node[v] (t8) at (1,-3) {};
    \node[v] (t9) at (1.5,-2) {};
    %%
    \draw (r)--(t1)--(t2)--(t3)--(t4);
    \draw (t1)--(t5);
    \draw (r)--(t6)--(t7)--(t8);
    \draw (t6)--(t9);       
    \end{scope}    
\end{tikzpicture}
\]
Construct the inverse of this map for the following binary trees.
  \[
\begin{tikzpicture}[scale=2,xscale=1.5]
 \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, minimum
   size=1.1mm]
    \begin{scope}[shift={(0,.7)},scale=.4]
    \node[v] (r) at (0,0) {};
    \node[v] (t1) at (-1,-1) {};
    \node[v] (t2) at (-1.5,-2) {};
    \node[v] (t3) at (-1.75,-3) {};
    \node[v] (t4) at (-.75,-3) {};
    \node[v] (t5) at (-.5,-2) {};
    \node[v] (t6) at (1,-1) {};
    \node[v] (t7) at (-1.25,-3) {};
    \node[v] (t8) at (1.5,-2) {};
    \node[v] (t9) at (-.25,-3) {};
    %%
    \draw (r)--(t1)--(t2)--(t3); \draw (t2)--(t7);
    \draw (t5)--(t4);
    \draw (t1)--(t5);
    \draw (r)--(t6); %\draw (t7)--(t8);
    \draw (t6)--(t8); \draw (t5)--(t9);
    \end{scope}    
    %%
    \begin{scope}[shift={(2.5,.7)},scale=.4]
    \node[v] (r) at (0,0) {};
    \node[v] (t1) at (-1,-1) {};
    \node[v] (t2) at (-1.5,-2) {};
    \node[v] (t3) at (1,-1) {};
    \node[v] (t4) at (1.5,-2) {};
    \node[v] (t5) at (1.75,-3) {};
    \node[v] (t6) at (1.825,-4) {};
    \node[v] (t7) at (1.625,-4) {};
    \node[v] (t8) at (1.25,-3) {};
    \node[v] (t9) at (.5,-2) {};
    %%
    \draw (r)--(t1)--(t2); \draw (r)--(t3)--(t4)--(t5)--(t6);
    \draw (t5)--(t7);
    \draw (t8)--(t4);
    \draw (t3)--(t9);
    \end{scope}
\end{tikzpicture}
\]

\item Recall the bijection of de Bruijn and Marselt from plane trees on $n+1$ vertices to binary trees on $n$ vertices; an example is shown below. 
  \[
  \begin{tikzpicture}[scale=1]
    \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, minimum size=1.1mm]
    %%
    \begin{scope}[shift={(0,0)},xscale=1.35]
      \node[v] (r) at (0,0) {};
      \node[v,label=left:{$a$}] (a) at (-1,-1) {};
      \node[v,label=left:{$b$}] (b) at (0,-1) {};
      \node[v,label=right:{$c$}] (c) at (1,-1) {};
      \node[v,label=left:{$d$}] (d) at (-1,-2) {};
      \node[v,label=below:{$e$}] (e) at (-.5,-2) {};
      \node[v,label=left:{$f$}] (f) at (0,-2) {};
      \node[v,label=below:{$g$}] (g) at (.5,-2) {};
      \node[v,label=below:{$h$}] (h) at (-1.5,-3) {};
      \node[v,label=below:{$i$}] (i) at (-.5,-3) {};
      \node[v,label=below:{$j$}] (j) at (0,-3) {};
      %%
      \draw (r)--(a)--(d)--(h); \draw (d)--(i);
      \draw (r)--(b)--(f)--(j);
      \draw (e)--(b)--(g); \draw (r)--(c);
    \end{scope}
    %%
    \begin{scope}[shift={(5.25,0)},xscale=1.35]
    \node[v,label=left:{$a$}] (a) at (-1,-1) {};
    \node[v,label=above:{$b$}] (b) at (0,-1) {};
    \node[v,label=above:{$c$}] (c) at (1,-1) {};
    \node[v,label=left:{$d$}] (d) at (-1,-2) {};
    \node[v,label=below:{$e$}] (e) at (-.5,-2) {};
    \node[v,label=above:{$f$}] (f) at (0,-2) {};
    \node[v,label=below:{$g$}] (g) at (.5,-2) {};
    \node[v,label=below:{$h$}] (h) at (-1.5,-3) {};
    \node[v,label=below:{$i$}] (i) at (-.5,-3) {};
    \node[v,label=below:{$j$}] (j) at (0,-3) {};
    %%
    \draw (a)--(d)--(h); \draw (b)--(e);  \draw (f)--(j);
    \draw[dashed] (a)--(b)--(c); \draw[dashed] (e)--(f)--(g); 
    \draw[dashed] (h)--(i);
   \end{scope}
    %%
   \begin{scope}[shift={(10,0)},scale=.75]
    \node[v,label=left:{$a$}] (a) at (0,0) {};
    \node[v,label=right:{$b$}] (b) at (1,-1) {};
    \node[v,label=below:{$c$}] (c) at (2,-2) {};
    \node[v,label=left:{$d$}] (d) at (-1,-1) {};
    \node[v,label=right:{$e$}] (e) at (0,-2) {};
    \node[v,label=right:{$f$}] (f) at (1,-3) {};
    \node[v,label=below:{$g$}] (g) at (2,-4) {};
    \node[v,label=left:{$h$}] (h) at (-2,-2) {};
    \node[v,label=below:{$i$}] (i) at (-1,-3) {};
    \node[v,label=below:{$j$}] (j) at (0,-4) {};
    %%
    \draw (a)--(d)--(h)--(i); \draw (a)--(b)--(e)--(f)--(j);  
    \draw (b)--(c); \draw (f)--(g);
   \end{scope}
\end{tikzpicture}
\]   
Construct the inverse of this bijection for the same two binary trees from the previous problem. \\

\end{enumerate}


  

\end{document}






  


