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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due Decmeber 8 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

%$\;$

\noindent\textbf{Topics}: Tutte polynomial, M\"obius functions


\begin{enumerate}


\item In class, we computed the Tutte polynomial recursively for the graph $K_4$ minus an edge, as follows:
  \[
\begin{tikzpicture}[scale=.5,
  v/.style={circle, fill, inner sep=1.15pt},
  %>= Stealth,
  line width=.7pt   % thicker arrows
]
%%----------------------------
\begin{scope}[shift={(0,0)}]
\node[v] (b) at (90:1) {}; 
\node[v] (a) at (180:1) {}; \node[v] (c) at (0:1) {}; 
\node[v] (d) at (270:1) {}; 
\draw (a) to (b); \draw (b) to (c); 
\draw(c) to (d); \draw (d) to (a); 
\draw[red,very thick] (b) to (d); 
\draw[gray,->] (-1,-1) to (-6,-2.5);
\draw[gray,->] (1,-1) to (6,-2.5);
\node at (9,0) {$T_{K_4{\setminus}e}(x,y)=x^3+2x^2+2xy+y^2+x+y$};
\end{scope}
%%============================
\begin{scope}[shift={(-8,-3.5)}]
\node[v] (b) at (90:1) {}; 
\node[v] (a) at (180:1) {}; \node[v] (c) at (0:1) {}; 
\node[v] (d) at (270:1) {}; 
\draw (a) to (b); \draw (b) to (c); 
\draw(c) to (d); \draw[red,very thick] (d) to (a); 
\draw[gray,->] (-1,-1) to (-3,-2.5);
\draw[gray,->] (1,-1) to (3,-2.5);
\end{scope}
%%----------------------------
\begin{scope}[shift={(7.75,-3.5)}]
\node[v] (b) at (0:0) {}; 
\node[v] (a) at (180:1) {}; \node[v] (c) at (0:1) {}; 
\draw[red,very thick] (a) to[bend left=50] (b); \draw (b) to[bend left=50] (c); 
\draw (a) to[bend right=50] (b); \draw (b) to[bend right=50] (c); 
\draw[gray,->] (-1,-1) to (-3,-2.5);
\draw[gray,->] (1,-1) to (3,-2.5);
\end{scope}
%%============================
\begin{scope}[shift={(-12,-7)}]
\node[v] (b) at (90:1) {}; 
\node[v] (a) at (180:1) {}; \node[v] (c) at (0:1) {}; 
\node[v] (d) at (270:1) {}; 
\draw (a) to (b); \draw (b) to (c); 
\draw(c) to (d); 
\node at (270:1.6) {$x^3$};
\end{scope}
%%----------------------------
\begin{scope}[shift={(-4,-7)}]
\node[v] (a) at (90:1) {}; 
\node[v] (b) at (210:1) {}; \node[v] (c) at (330:1) {}; 
\draw (a) to (b); \draw[very thick,red] (b) to (c); 
\draw (c) to (a); 
\draw[gray,->] (-1,-1) to (-2,-2);
\draw[gray,->] (1,-1) to (2,-2);
\end{scope}
%%----------------------------
\begin{scope}[shift={(4,-7)}]
\node[v] (b) at (0:0) {}; 
\node[v] (a) at (180:1) {}; \node[v] (c) at (0:1) {}; 
\draw[red,very thick] (a) to (b); \draw (b) to[bend left=50] (c); 
\draw (b) to[bend right=50] (c); 
\draw[gray,->] (-1,-1) to (-2,-2.5);
\draw[gray,->] (1,-1) to (2,-2.5);
\end{scope}
%%----------------------------
\begin{scope}[shift={(12.25,-7)}]
\node[v] (b) at (0:.5) {}; 
\node[v] (a) at (180:.5) {}; 
\draw[looseness=20, out=150, in=210] (a) to (a);
\draw[red,very thick] (a) to[bend right=50] (b); 
\draw (a) to[bend left=50] (b); 
\draw[gray,->] (-1,-1) to (-2,-2.5);
\draw[gray,->] (1,-1) to (2,-2.5);
\end{scope}
%%============================
\begin{scope}[shift={(-6.25,-10.5)}]
\node[v] (a) at (90:1) {}; 
\node[v] (b) at (210:1) {}; \node[v] (c) at (330:1) {}; 
\draw (a) to (b); \draw (c) to (a); 
\node at (270:1.3) {$x^2$};
\end{scope}
%%----------------------------
\begin{scope}[shift={(-1.75,-10.5)}]
\node[v] (a) at (90:.75) {}; 
\node[v] (b) at (270:.75) {};
\draw[very thick,red] (a) to[bend right=50] (b); 
\draw[very thick] (a) to[bend left=50] (b); 
\draw[gray,->] (-.5,-1) to (-1.25,-2);
\draw[gray,->] (.5,-1) to (1.25,-2);
\end{scope}
%%----------------------------
\begin{scope}[shift={(1.75,-10.5)}]
\node[v] (a) at (180:.75) {}; 
\node[v] (b) at (0:0) {};
\node[v] (c) at (0:.75) {};
\draw (a) to (b); \draw (b) to (c); 
\node at (270:1.3) {$x^2$};
\end{scope}
%%----------------------------
\begin{scope}[shift={(6.25,-10.5)}]
\node[v] (a) at (180:.5) {}; 
\node[v] (b) at (0:.5) {};
\draw (a) to (b); 
\draw[looseness=20, out=30, in=-30] (b) to (b);
\node at (270:1.3) {$xy$};
\end{scope}
%%----------------------------
\begin{scope}[shift={(10.5,-10.5)}]
\node[v] (a) at (180:.5) {}; 
\node[v] (b) at (0:.5) {};
\draw (a) to (b); 
\draw[looseness=20, out=150, in=210] (a) to (a);
\node at (270:1.3) {$xy$};
\end{scope}
%%----------------------------
\begin{scope}[shift={(14.25,-10.5)}]
\node[v] (a) at (0:0) {}; 
\draw[looseness=20, out=30, in=-30] (a) to (a);
\draw[looseness=20, out=150, in=210] (a) to (a);
\node at (270:1.3) {$y^2$};
\end{scope}
%%============================
\begin{scope}[shift={(-3.25,-14)}]
\node[v] (a) at (90:.75) {}; 
\node[v] (b) at (270:.75) {};
\draw (a) to (b); 
\node at (270:1.5) {$x$};
\end{scope}
%%----------------------------
\begin{scope}[shift={(-.5,-14)}]
\node[v] (a) at (0:0) {}; 
\draw[looseness=20, out=30, in=-30] (a) to (a);
\node at (270:1.5) {$y$};
\end{scope}
\end{tikzpicture}
\]
\begin{enumerate}
\item Compute the Tutte polynomial $T_G(x,y)$, where $G$ the ``house
  graph'' from HW 8.
\item Find a graph $G_1$ such that $T_{G_1}(x,y)=xT_G(x,y)$.
\item Find a graph $G_2$ such that $T_{G_2}(x,y)=yT_G(x,y)$. \\
\end{enumerate}



\item The \emph{M\"obius function} of a poset is defined 
\[
\mu\colon\Int(P)\longto\R,\qquad\mu(x,y)=\begin{cases}1 & x=y \\
-\Sum_{z\leq x}\mu(z,x) & \text{otherwise}.
\end{cases}
\]
A Hasse diagram of the partition lattice $\Pi_4$ is shown below, along with its nodes labeled by $\mu(\hat{0},\pi)$, where $\hat{0}=1/2/3/4$ is the minimum element.
\[
\hspace*{-3mm}
\begin{tikzpicture}[scale=1.25,every node/.style={font=\scriptsize}]
 \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, 
    minimum size=.75mm]
%%
    \begin{scope}[shift={(0,0)}]
\node (1234) at (0,3) {$1234$};
%%
\node (14-23) at (-3,2) {$14/23$};
\node (1-234) at (-2,2) {$1/234$}; 
\node (124-3) at (-1,2) {$124/3$}; 
\node (13-24) at (0,2) {$13/24$}; 
\node (123-4) at (1,2) {$123/4$}; 
\node (134-2) at (2,2) {$134/2$}; 
\node (12-34) at (3,2) {$12/34$}; 
%%
\node (1-23-4) at (-2.5,1) {$1/23/4$};
\node (14-2-3) at (-1.5,1) {$14/2/3$}; 
\node (1-24-3) at (-.5,1) {$1/24/3$}; 
\node (13-2-4) at (.5,1) {$13/2/4$}; 
\node (12-3-4) at (1.5,1) {$12/3/4$}; 
\node (1-2-34) at (2.5,1) {$1/2/34$}; 
\node (1-2-3-4) at (0,0) {$1/2/3/4$}; 

    \draw (1234)--(14-23);
    \draw (1234)--(1-234);
    \draw (1234)--(123-4);
    \draw (1234)--(124-3);
    \draw (1234)--(134-2);
    \draw (1234)--(12-34);
    \draw (1234)--(13-24);
    
    %%
    \draw (14-23)--(1-23-4);
    \draw (14-23)--(14-2-3);
    \draw (1-234)--(1-23-4);
    \draw (1-234)--(1-2-34);
    \draw (1-234)--(1-24-3);
    \draw (124-3)--(12-3-4);
    \draw (124-3)--(14-2-3);
    \draw (124-3)--(1-24-3);
    \draw (123-4)--(12-3-4);
    \draw (123-4)--(13-2-4);
    \draw (123-4)--(1-23-4);
    \draw (134-2)--(13-2-4);
    \draw (134-2)--(14-2-3);
    \draw (134-2)--(1-2-34);
    \draw (12-34)--(12-3-4);
    \draw (12-34)--(1-2-34);
    \draw (13-24)--(1-24-3);
    \draw (13-24)--(13-2-4);
    %%
    \draw (1-2-3-4)--(1-23-4);
    \draw (1-2-3-4)--(14-2-3);
    \draw (1-2-3-4)--(1-24-3);
    \draw (1-2-3-4)--(13-2-4);
    \draw (1-2-3-4)--(12-3-4);
    \draw (1-2-3-4)--(1-2-34);
    \end{scope}    
    %%
    \begin{scope}[shift={(6.5,0)},xscale=.9,every node/.style={font=\small}]
\node (1234) at (0,3) {$-6$};
%%
\node (14-23) at (-3,2) {$1$};
\node (1-234) at (-2,2) {$2$}; 
\node (124-3) at (-1,2) {$2$}; 
\node (13-24) at (0,2) {$1$}; 
\node (123-4) at (1,2) {$2$}; 
\node (134-2) at (2,2) {$2$}; 
\node (12-34) at (3,2) {$1$}; 
%%
\node (1-23-4) at (-2.5,1) {$-1$};
\node (14-2-3) at (-1.5,1) {$-1$};
\node (1-24-3) at (-.5,1) {$-1$};
\node (13-2-4) at (.5,1) {$-1$};
\node (12-3-4) at (1.5,1) {$-1$};
\node (1-2-34) at (2.5,1) {$-1$};
\node (1-2-3-4) at (0,0) {$1$}; 

    \draw (1234)--(14-23);
    \draw (1234)--(1-234);
    \draw (1234)--(123-4);
    \draw (1234)--(124-3);
    \draw (1234)--(134-2);
    \draw (1234)--(12-34);
    \draw (1234)--(13-24);
    
    %%
    \draw (14-23)--(1-23-4);
    \draw (14-23)--(14-2-3);
    \draw (1-234)--(1-23-4);
    \draw (1-234)--(1-2-34);
    \draw (1-234)--(1-24-3);
    \draw (124-3)--(12-3-4);
    \draw (124-3)--(14-2-3);
    \draw (124-3)--(1-24-3);
    \draw (123-4)--(12-3-4);
    \draw (123-4)--(13-2-4);
    \draw (123-4)--(1-23-4);
    \draw (134-2)--(13-2-4);
    \draw (134-2)--(14-2-3);
    \draw (134-2)--(1-2-34);
    \draw (12-34)--(12-3-4);
    \draw (12-34)--(1-2-34);
    \draw (13-24)--(1-24-3);
    \draw (13-24)--(13-2-4);
    %%
    \draw (1-2-3-4)--(1-23-4);
    \draw (1-2-3-4)--(14-2-3);
    \draw (1-2-3-4)--(1-24-3);
    \draw (1-2-3-4)--(13-2-4);
    \draw (1-2-3-4)--(12-3-4);
    \draw (1-2-3-4)--(1-2-34);
    \end{scope}    
    %%
    
\end{tikzpicture}
\]
The M\"obius function of $\Pi_4$ can be represented in matrix form, as follows.
\[
\begin{array}{c}
1/2/3/4 \\
1/23/4 \\
14/2/3 \\
1/24/3 \\
13/2/4 \\
12/3/4 \\
1/2/34 \\ %
14/23 \\
1/234 \\
124/3 \\
13/24 \\
123/4 \\
134/2 \\
12/34 \\
1234
\end{array}
\left[
\begin{array}{ccccccccccccccc}
     1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & -6 \\
     0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & 0 & 0 & 2 \\
     0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & -1 & 0 & 2 \\
     0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 0 & 0 & 0 & 2 \\
     0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 0 & 2 \\
     0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & 2 \\
     0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & -1 & -1 & 2 \\ %%-----
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \\
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 \\
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \\
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 \\
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 \\
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\ %%---
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right]
\]
Create an analogous diagram and matrix for the M\"obius function of the Tamari lattice.
\[
\begin{tikzpicture}[scale=2.25]
    \begin{scope}[shift={(0,0)},every node/.style={font=\footnotesize}]
    \node(4-5) at (.5,4.5) {$a(b(c(de))$};
    \node(3-5L) at (.5,3.5) {$a(b((cd)e))$};
    \node(3-5R) at (1.5,3.5) {$(ab)(c(de))$};
    \node(3L) at (-1,3) {$a((bc)(de))$};
    \node(3R) at (0,3) {$a((b(cd))e)$}; 
    \node(2-5L) at (-.5,2.5) {$a(((bc)d)e)$};
    \node(2-5R) at (1.5,2.5) {$(ab)((cd)e)$};
    \node(2L) at (-1,2) {$(a(bc))(de)$}; 
    \node(2R) at (1,2) {$(a(b(cd)))e$};
    \node(1-5L) at (.5,1.5) {$(a((bc)d))e$};
    \node(1-5R) at (1.5,1.5) {$((ab)(cd))e$};
    \node(1L) at (-1,1) {$((ab)c)(de)$};
    \node(1R) at (0,1) {$((a(bc))d)e$};
    \node(0) at (0,0) {$(((ab)c)d)e$};
    %%
    \draw (0)--(1L); \draw (0)--(1R); \draw (0)--(1-5R);
    \draw (1L)--(2L); \draw (1L)--(3-5R);
    \draw (1R)--(2L); \draw (1R)--(1-5L);
    \draw (1-5L)--(2-5L); \draw (1-5L)--(2R);
    \draw (1-5R)--(2R); \draw (1-5R)--(2-5R);
    \draw (2L)--(3L); \draw (2R)--(3R); 
    \draw (2-5L)--(3L); \draw (2-5L)--(3R);
    \draw (2-5R)--(3-5L); \draw (2-5R)--(3-5R);
    \draw (3L)--(4-5); \draw (3R)--(3-5L);
    \draw (3-5L)--(4-5); \draw (3-5R)--(4-5);   
    \end{scope}
    %%
%    \begin{scope}[shift={(3.5,0)},every node/.style={font=\footnotesize}]
%    \node(4-5) at (.5,4.5) {}; %{$(a(b(c(de)))$};
%    \node(3-5L) at (.5,3.5) {}; %{$a(b((cd)e))$};
%    \node(3-5R) at (1.5,3.5) {}; %{$(ab)(c(de))$};
%    \node(3L) at (-1,3) {}; %{$a((bc)(de))$};
%    \node(3R) at (0,3) {}; %{$a((b(cd))e)$}; 
%    \node(2-5L) at (-.5,2.5) {}; %{$a(((bc)d)e)$};
%    \node(2-5R) at (1.5,2.5) {}; %{$(ab)((cd)e)$};
%    \node(2L) at (-1,2) {}; %{$(a(bc))(de)$}; 
%    \node(2R) at (1,2) {}; %{$(a(b(cd)))e$};
%    \node(1-5L) at (.5,1.5) {}; %{$(a((bc)d))e$};
%    \node(1-5R) at (1.5,1.5) {}; %{$((ab)(cd))e$};
%    \node(1L) at (-1,1) {}; %{$((ab)c)(de)$};
%    \node(1R) at (0,1) {}; %{$((a(bc))d)e$};
%    \node(0) at (0,0) {}; %{$(((ab)c)d)e$};
%    %%
%    \draw (0)--(1L); \draw (0)--(1R); \draw (0)--(1-5R);
%    \draw (1L)--(2L); \draw (1L)--(3-5R);
%    \draw (1R)--(2L); \draw (1R)--(1-5L);
%    \draw (1-5L)--(2-5L); \draw (1-5L)--(2R);
%    \draw (1-5R)--(2R); \draw (1-5R)--(2-5R);
%    \draw (2L)--(3L); \draw (2R)--(3R); 
%    \draw (2-5L)--(3L); \draw (2-5L)--(3R);
%    \draw (2-5R)--(3-5L); \draw (2-5R)--(3-5R);
%    \draw (3L)--(4-5); \draw (3R)--(3-5L);
%    \draw (3-5L)--(4-5); \draw (3-5R)--(4-5);   
%    \end{scope}
\end{tikzpicture}
\]

\item Let $\mathcal{F}_n$ be the set of labeled forests on $[n]$. To every forest $F\in\mathcal{F}_n$, we can associate a partition $\pi\in\Pi_n$ defined by the connected components; write this as $c(F)=\pi$. Set up a M\"obius inversion problem by defining two functions $f,g\colon\Pi_n\to\Z$, where an evaluation of one of them gives Cayley's formula for the number of labeled trees. Explain your steps, but you do \emph{not} need to solve it. 


\end{enumerate}

\end{document}
