\documentclass[multi]{standalone}
\standaloneenv{page}

\usepackage{amsmath,amssymb}
\usepackage{mathpazo}
\usepackage{ifthen}
\usepackage{tikz,pgf,pgfmath}
	\usetikzlibrary{shapes,calc,decorations.pathmorphing}
	\pgfdeclarelayer{background}
	\pgfdeclarelayer{middle}
	\pgfsetlayers{background,middle,main}
	\tikzstyle{fnode}=[fill=black,draw=black,circle,scale=\s]
% 	\tikzstyle{back}=[loosely dotted, thin]
	\tikzstyle{back}=[thin,dashed]
	\tikzstyle{edge}=[line width=.75pt]
	\newcommand{\red}{red!50!white}
	\newcommand{\blue}{blue!50!white}
	\newcommand{\redLabel}{red!50!black}
	\newcommand{\blueLabel}{blue!50!black}
	\newcommand{\green}{green!50!gray}
	\newcommand{\gray}{white!50!gray}
	\tikzstyle{facet}=[fill=\green,fill opacity=0.6]
	\tikzstyle{facetR}=[fill=\red,fill opacity=0.8]
	\tikzstyle{facetL}=[fill=\blue,fill opacity=0.8]
	\tikzstyle{pathnode}=[inner sep=.9pt]
%some counters for later
\newcounter{i}
\newcounter{n}
\newcounter{m}
\newcounter{curx}
\newcounter{cury}
\newcounter{orient}
%%a macro to draw nu-trees
\newcommand{\nuTree}[7]{
	%Parameter 1: bounding path!!!always starts with a north step
	%Parameter 2: nodes
	%Parameter 3: edges
	%Parameter 4: special coordinates
	%Parameter 5: scaling
	%Parameter 6: text
	%Parameter 7: background color
	%%
	\begin{tikzpicture}
		\def\dx{#5};
		\def\s{3*\dx};	%needed for scaling the nodes correctly; see fnode-style
		%%
		\setcounter{n}{0}
		\setcounter{m}{0}
		\setcounter{orient}{0}
		\setcounter{curx}{0}
		\setcounter{cury}{0}
		%%compute y-coordinate of endpoint of nu
		\foreach \step in {#1}{
			\ifthenelse{\equal{\theorient}{0}}{
				\addtocounter{n}{\step}
				\setcounter{orient}{1}
			}{
				\addtocounter{m}{\step}
				\setcounter{orient}{0}
			}
		}
		%%draw nu and background grid
		\setcounter{i}{0}
		\setcounter{orient}{0}
		\foreach \step in {#1}{
			\ifthenelse{\equal{\theorient}{0}}{
				\coordinate(q\thei) at (\thecurx*\dx,\thecury*\dx);
				\stepcounter{i}
				\coordinate(q\thei) at (\thecurx*\dx,{(\thecury+\step)*\dx});
				\stepcounter{i}
				\draw[\gray,dashed](\thecurx*\dx,\thecury*\dx) -- (\thecurx*\dx,{(\thecury+\step)*\dx});
				\begin{pgfonlayer}{background}
					\foreach \k in {1,...,\step}{
						\draw[\gray,dashed](0*\dx,{(\thecury+\k)*\dx}) -- (\thecurx*\dx,{(\thecury+\k)*\dx});
					}
				\end{pgfonlayer}
				\addtocounter{cury}{\step}
				\setcounter{orient}{1}
			}{
				\draw[\gray,dashed](\thecurx*\dx,\thecury*\dx) -- ({(\thecurx+\step)*\dx},\thecury*\dx);
				\begin{pgfonlayer}{background}
					\foreach \k in {0,...,\step}{
						\ifthenelse{\equal{\k}{\step}}{
						}{
							\draw[\gray,dashed]({(\thecurx+\k)*\dx},\thecury*\dx) -- ({(\thecurx+\k)*\dx},\then*\dx);
						}
					}
				\end{pgfonlayer}
				\addtocounter{curx}{\step}
				\setcounter{orient}{0}
			}
		}
		%%draw background color
		\ifthenelse{\equal{#7}{}}{}{
			\coordinate(zero) at (0*\dx,0*\dx);
			\coordinate(q\thei) at(0*\dx,\then*\dx);
			\addtocounter{i}{-1}
			\foreach \k in {0,2,...,\thei}{
				\pgfmathparse{\k+1}
				\let\res\pgfmathresult
				\begin{pgfonlayer}{middle}
					\fill[#7] (zero |- q\k) -- (q\k) -- (q\res) -- (zero |- q\res) -- cycle;
				\end{pgfonlayer}
			}
		}
		%%draw nodes
		\setcounter{i}{1}
		\foreach \a/\b/\c/\d in {#2}{
			\draw(\a*\dx,\b*\dx) node[fnode,draw=\c,fill=\c](p\thei){};
			\ifthenelse{\equal{\d}{d}}{
				\draw[orange!50!gray,decorate,decoration={snake,amplitude=.5,segment length=2.5}](p\thei) -- (\a*\dx,{(\then+.5)*\dx});
			}{
			}
			\stepcounter{i}
		}
		%%draw edges
		\foreach \a/\b in {#3}{
			\draw[line width=3*\dx pt](\a) -- (\b);
		}
 		%%draw special coordinates
 		\foreach \a/\b/\c in {#4}{
			\draw[\c,line width=.5pt](\a*\dx,\b*\dx) circle(.25*\dx);
 		}
		%%draw text label
		\foreach \a/\b/\c/\d in {#6}{
			\draw[\d](\a*\dx,\b*\dx) node[scale=.75,anchor=west]{\c};
		}
	\end{tikzpicture}
}
%a macro to draw nu-paths
\newcommand{\nuPath}[6]{
	%Parameter 1: bounding path!!!always starts with a north step
	%Parameter 2: height vector
	%Parameter 3: marked coordinates
	%Parameter 4: scaling
	%Parameter 5: text
	%Parameter 6: background color
	%%
	\begin{tikzpicture}
		\def\dx{#4};
		%%
		\setcounter{n}{0}
		\setcounter{m}{0}
		\setcounter{orient}{0}
		\setcounter{curx}{0}
		\setcounter{cury}{0}
		%%compute y-coordinate of endpoint of nu
		\foreach \step in {#1}{
			\ifthenelse{\equal{\theorient}{0}}{
				\addtocounter{n}{\step}
				\setcounter{orient}{1}
			}{
				\addtocounter{m}{\step}
				\setcounter{orient}{0}
			}
		}
		%%draw nu and background grid
		\setcounter{i}{0}
		\setcounter{orient}{0}
		\foreach \step in {#1}{
			\ifthenelse{\equal{\theorient}{0}}{
				\coordinate(q\thei) at (\thecurx*\dx,\thecury*\dx);
				\stepcounter{i}
				\coordinate(q\thei) at (\thecurx*\dx,{(\thecury+\step)*\dx});
				\stepcounter{i}
				\draw[\gray,dashed](\thecurx*\dx,\thecury*\dx) -- (\thecurx*\dx,{(\thecury+\step)*\dx});
				\begin{pgfonlayer}{background}
					\foreach \k in {1,...,\step}{
						\draw[\gray,dashed](0*\dx,{(\thecury+\k)*\dx}) -- (\thecurx*\dx,{(\thecury+\k)*\dx});
					}
				\end{pgfonlayer}
				\addtocounter{cury}{\step}
				\setcounter{orient}{1}
			}{
				\draw[\gray,dashed](\thecurx*\dx,\thecury*\dx) -- ({(\thecurx+\step)*\dx},\thecury*\dx);
				\begin{pgfonlayer}{background}
					\foreach \k in {0,...,\step}{
						\ifthenelse{\equal{\k}{\step}}{
						}{
							\draw[\gray,dashed]({(\thecurx+\k)*\dx},\thecury*\dx) -- ({(\thecurx+\k)*\dx},\then*\dx);
						}
					}
				\end{pgfonlayer}
				\addtocounter{curx}{\step}
				\setcounter{orient}{0}
			}
		}
		%%draw background color
		\ifthenelse{\equal{#6}{}}{}{
			\coordinate(zero) at (0*\dx,0*\dx);
			\coordinate(q\thei) at(0*\dx,\then*\dx);
			\addtocounter{i}{-1}
			\foreach \k in {0,2,...,\thei}{
				\pgfmathparse{\k+1}
				\let\res\pgfmathresult
				\begin{pgfonlayer}{background}
					\fill[#6] (zero |- q\k) -- (q\k) -- (q\res) -- (zero |- q\res) -- cycle;
				\end{pgfonlayer}
			}
		}
		%%draw path
		\setcounter{i}{0}
		\setcounter{cury}{0}
 		\foreach \a in {#2}{
			\ifthenelse{\equal{\thei}{0}}{
				\draw[line width=2pt,rounded corners](\thei*\dx,\thei*\dx) -- (\thei*\dx,{\thei+\a)*\dx});
			}{
				\draw[line width=2pt,rounded corners]({(\thei-1)*\dx},\thecury*\dx) -- (\thei*\dx,\thecury*\dx);
				\draw[line width=2pt,rounded corners](\thei*\dx,\thecury*\dx) -- (\thei*\dx,\a*\dx);
			}
			\setcounter{cury}{\a}
			\stepcounter{i}
 		}
 		%%draw special coordinates
 		\foreach \a/\b/\c/\d in {#3}{
			\ifthenelse{\equal{\c}{}}{}{
				\draw[\c,line width=.5pt](\a*\dx,\b*\dx) circle(.25*\dx);
			}
			\fill[\d,line width=.5pt](\a*\dx,\b*\dx) circle(.15*\dx);
 		}
		%%draw text label
		\foreach \a/\b/\c/\d in {#5}{
			\draw[\d](\a*\dx,\b*\dx) node[scale=.9,anchor=west]{\c};
		}
	\end{tikzpicture}
}

\begin{document}

%Figure: Northeast path with Ferrers shape.
\begin{page}%page 1
	\begin{tikzpicture}\small
		\def\x{.75};
		%%
		\draw[line width=1pt](0*\x,0*\x) -- (1*\x,0*\x) -- (1*\x,1*\x) -- (3*\x,1*\x) -- (3*\x,3*\x) -- (5*\x,3*\x) -- (5*\x,4*\x);% -- (6*\x,4*\x);
		%%
		\draw[dashed](0*\x,1*\x) -- (1*\x,1*\x);
		\draw[dashed](0*\x,2*\x) -- (3*\x,2*\x);
		\draw[dashed](0*\x,3*\x) -- (3*\x,3*\x);
		\draw[dashed](0*\x,4*\x) -- (5*\x,4*\x);
		\draw[dashed](0*\x,0*\x) -- (0*\x,4*\x);
		\draw[dashed](1*\x,1*\x) -- (1*\x,4*\x);
		\draw[dashed](2*\x,1*\x) -- (2*\x,4*\x);
		\draw[dashed](3*\x,3*\x) -- (3*\x,4*\x);
		\draw[dashed](4*\x,3*\x) -- (4*\x,4*\x);
		%%
		\draw(0*\x,0*\x) node[fill,circle,scale=.4]{};
		\draw(0*\x,1*\x) node[fill,circle,scale=.4]{};
		\draw(0*\x,2*\x) node[fill,circle,scale=.4]{};
		\draw(0*\x,3*\x) node[fill,circle,scale=.4]{};
		\draw(0*\x,4*\x) node[fill,circle,scale=.4]{};
		\draw(1*\x,0*\x) node[fill,circle,scale=.4]{};
		\draw(1*\x,1*\x) node[fill,circle,scale=.4]{};
		\draw(1*\x,2*\x) node[fill,circle,scale=.4]{};
		\draw(1*\x,3*\x) node[fill,circle,scale=.4]{};
		\draw(1*\x,4*\x) node[fill,circle,scale=.4]{};
		\draw(2*\x,1*\x) node[fill,circle,scale=.4]{};
		\draw(2*\x,2*\x) node[fill,circle,scale=.4]{};
		\draw(2*\x,3*\x) node[fill,circle,scale=.4]{};
		\draw(2*\x,4*\x) node[fill,circle,scale=.4]{};
		\draw(3*\x,1*\x) node[fill,circle,scale=.4]{};
		\draw(3*\x,2*\x) node[fill,circle,scale=.4]{};
		\draw(3*\x,3*\x) node[fill,circle,scale=.4]{};
		\draw(3*\x,4*\x) node[fill,circle,scale=.4]{};
		\draw(4*\x,3*\x) node[fill,circle,scale=.4]{};
		\draw(4*\x,4*\x) node[fill,circle,scale=.4]{};
		\draw(5*\x,3*\x) node[fill,circle,scale=.4]{};
		\draw(5*\x,4*\x) node[fill,circle,scale=.4]{};
		\begin{pgfonlayer}{background}
			\fill[\gray,opacity=.5](0*\x,0*\x) -- (5*\x,0*\x) -- (5*\x,4*\x) -- (0*\x,4*\x) -- cycle;
		\end{pgfonlayer}
	\end{tikzpicture}
\end{page}

%Figure: Illustration of $\nu$-rotation
\begin{page}%page 2
	\begin{tikzpicture}
		\draw(1,1) node{\begin{tikzpicture}\small
			\def\x{.75};
			%%
			\draw[dashed,\gray](0*\x,0*\x) -- (1*\x,0*\x);
			\draw[dashed,\gray](0*\x,1*\x) -- (3*\x,1*\x);
			\draw[dashed,\gray](0*\x,2*\x) -- (3*\x,2*\x);
			\draw[dashed,\gray](0*\x,3*\x) -- (5*\x,3*\x);
			\draw[dashed,\gray](0*\x,4*\x) -- (5*\x,4*\x);
			\draw[dashed,\gray](0*\x,0*\x) -- (0*\x,4*\x);
			\draw[dashed,\gray](1*\x,0*\x) -- (1*\x,4*\x);
			\draw[dashed,\gray](2*\x,1*\x) -- (2*\x,4*\x);
			\draw[dashed,\gray](3*\x,1*\x) -- (3*\x,4*\x);
			\draw[dashed,\gray](4*\x,3*\x) -- (4*\x,4*\x);
			\draw[dashed,\gray](5*\x,3*\x) -- (5*\x,4*\x);
			%%
			\draw[line width=1.5pt,black](0*\x,0*\x) -- (0*\x,2*\x) -- (1*\x,2*\x);
			\draw[line width=1.5pt,black](4*\x,4*\x) -- (5*\x,4*\x);
			%%
			\draw(2*\x,2*\x) node[fill=\blue,circle,scale=.6](v){};
			\draw(4*\x,4*\x) node[fill=\red,circle,scale=.6](w){};
			\draw[line width=1pt,\blue,decorate,decoration={coil,amplitude=.5,segment length=5}](1*\x,2*\x) -- (2*\x,2*\x);
			\draw[line width=1.5pt,\red](v) -- (2*\x,3*\x) -- (3*\x,3*\x) -- (3*\x,4*\x) -- (w);
		\end{tikzpicture}};
		\draw(3.5,1) node{$\lessdot_{\nu}$};
		\draw(6,1) node{\begin{tikzpicture}\small
			\def\x{.75};
			%%
			\draw[dashed,\gray](0*\x,0*\x) -- (1*\x,0*\x);
			\draw[dashed,\gray](0*\x,1*\x) -- (3*\x,1*\x);
			\draw[dashed,\gray](0*\x,2*\x) -- (3*\x,2*\x);
			\draw[dashed,\gray](0*\x,3*\x) -- (5*\x,3*\x);
			\draw[dashed,\gray](0*\x,4*\x) -- (5*\x,4*\x);
			\draw[dashed,\gray](0*\x,0*\x) -- (0*\x,4*\x);
			\draw[dashed,\gray](1*\x,0*\x) -- (1*\x,4*\x);
			\draw[dashed,\gray](2*\x,1*\x) -- (2*\x,4*\x);
			\draw[dashed,\gray](3*\x,1*\x) -- (3*\x,4*\x);
			\draw[dashed,\gray](4*\x,3*\x) -- (4*\x,4*\x);
			\draw[dashed,\gray](5*\x,3*\x) -- (5*\x,4*\x);
			%%
			\draw[line width=1.5pt,black](0*\x,0*\x) -- (0*\x,2*\x) -- (1*\x,2*\x);
			\draw[line width=1.5pt,black](4*\x,4*\x) -- (5*\x,4*\x);
			%%
			\draw[line width=1pt,\blue,decorate,decoration={coil,amplitude=.5,segment length=5}](3*\x,4*\x) -- (4*\x,4*\x);
			\draw[line width=1.5pt,\red](1*\x,2*\x) -- (1*\x,3*\x) -- (2*\x,3*\x) -- (2*\x,3.98*\x) -- (3*\x,3.98*\x);
			%%
			\draw(2*\x,2*\x) node[draw=\blue,circle,scale=.6](v){};
			\draw(4*\x,4*\x) node[draw=\red,circle,scale=.6](w){};
		\end{tikzpicture}};
	\end{tikzpicture}
\end{page}

%Figure: The $\nu$-Tamari lattice for $\nu=EENEN$.
\begin{page}%page 3
	\begin{tikzpicture}\small
		\def\x{1};
		\def\y{.75};
		%%
		\draw(5*\x,1*\y) node[pathnode](n1){\nuPath{0,2,1,1,1}{0,0,1,2}{2/0/\redLabel/\blue,3/1/\redLabel/\blue}{.4}{}{}};
			\draw(4.25*\x,1.25*\y) node[\redLabel,anchor=east,scale=.75]{$\mathbf{(y{+}1)^{2}}$};
			\draw(4.25*\x,.75*\y) node[\blueLabel,anchor=east,scale=.75]{$\mathbf{x^{2}y^{2}}$};
		\draw(8*\x,4*\y) node[pathnode](n2){\nuPath{0,2,1,1,1}{0,1,1,2}{1/0//\blue,3/1/\redLabel/\blue}{.4}{}{}};
			\draw(8.75*\x,4.25*\y) node[\redLabel,anchor=west,scale=.75]{$\mathbf{(x{+}1)(y{+}1)}$};
			\draw(8.75*\x,3.75*\y) node[\blueLabel,anchor=west,scale=.75]{$\mathbf{x^{2}y}$};
		\draw(1*\x,5*\y) node[pathnode](n3){\nuPath{0,2,1,1,1}{0,0,2,2}{2/0/\redLabel/\blue}{.4}{}{}};
			\draw(.25*\x,5.25*\y) node[\redLabel,anchor=east,scale=.75]{$\mathbf{x(y{+}1)}$};
			\draw(.25*\x,4.75*\y) node[\blueLabel,anchor=east,scale=.75]{$\mathbf{xy}$};
		\draw(8*\x,6*\y) node[pathnode](n4){\nuPath{0,2,1,1,1}{0,1,2,2}{1/0//\blue,2/1//\blue}{.4}{}{}};
			\draw(7.25*\x,6.25*\y) node[\redLabel,anchor=east,scale=.75]{$\mathbf{(x{+}1)^{2}}$};
			\draw(7.25*\x,5.75*\y) node[\blueLabel,anchor=east,scale=.75]{$\mathbf{x^{2}}$};
		\draw(10*\x,6*\y) node[pathnode](n5){\nuPath{0,2,1,1,1}{1,1,1,2}{3/1/\redLabel/\blue}{.4}{}{}};
			\draw(10.75*\x,6.25*\y) node[\redLabel,anchor=west,scale=.75]{$\mathbf{x(y{+}1)}$};
			\draw(10.75*\x,5.75*\y) node[\blueLabel,anchor=west,scale=.75]{$\mathbf{xy}$};
		\draw(10*\x,8*\y) node[pathnode](n6){\nuPath{0,2,1,1,1}{1,1,2,2}{2/1//\blue}{.4}{}{}};
			\draw(10.75*\x,8.25*\y) node[\redLabel,anchor=west,scale=.75]{$\mathbf{x(x{+}1)}$};
			\draw(10.75*\x,7.75*\y) node[\blueLabel,anchor=west,scale=.75]{$\mathbf{x}$};
		\draw(5*\x,9*\y) node[pathnode](n7){\nuPath{0,2,1,1,1}{0,2,2,2}{1/0//\blue}{.4}{}{}};
			\draw(4.25*\x,9.25*\y) node[\redLabel,anchor=east,scale=.75]{$\mathbf{x(x{+}1)}$};
			\draw(4.25*\x,8.75*\y) node[\blueLabel,anchor=east,scale=.75]{$\mathbf{x}$};
		\draw(10*\x,10*\y) node[pathnode](n8){\nuPath{0,2,1,1,1}{1,2,2,2}{1/1//\blue}{.4}{}{}};
			\draw(10.75*\x,10.25*\y) node[\redLabel,anchor=west,scale=.75]{$\mathbf{x(x{+}1)}$};
			\draw(10.75*\x,9.75*\y) node[\blueLabel,anchor=west,scale=.75]{$\mathbf{x}$};
		\draw(8*\x,12*\y) node[pathnode](n9){\nuPath{0,2,1,1,1}{2,2,2,2}{}{.4}{}{}};
			\draw(7.25*\x,12.25*\y) node[\redLabel,anchor=east,scale=.75]{$\mathbf{x^{2}}$};
			\draw(7.25*\x,11.75*\y) node[\blueLabel,anchor=east,scale=.75]{$\mathbf{1}$};
		%%
		\draw(n1) -- (n2);
		\draw(n1) -- (n3);
		\draw(n2) -- (n4);
		\draw(n2) -- (n5);
		\draw(n3) -- (n7);
		\draw(n4) -- (n6);
		\draw(n4) -- (n7);
		\draw(n5) -- (n6);
		\draw(n6) -- (n8);
		\draw(n7) -- (n9);
		\draw(n8) -- (n9);
	\end{tikzpicture}
\end{page}

%Figure: A $\nu$-tree.
\begin{page}%page 4
	\begin{tikzpicture}
		\draw(1,1) node[pathnode]{\nuTree{1,3,2,5,3,1}{0/0//,1/1//,2/1//,3/1//,1/2//,6/3//,7/3//,8/3//,4/4//,5/4//,6/4//,0/5//,1/5//,4/5//,0/6//,9/6//}{p1/p12,p2/p3,p2/p5,p3/p4,p5/p13,p6/p7,p6/p11,p7/p8,p9/p10,p9/p14,p10/p11,p12/p15,p12/p13,p13/p14,p15/p16}{}{.6}{}{}};
	\end{tikzpicture}
\end{page}

%Figure: A binary tree.
\begin{page}%page 5
	\begin{tikzpicture}
		\def\x{.5};
		\def\s{.4};
		%%
		\draw(1*\x,10*\x) node[fnode](n1){};
		\draw(.25*\x,9*\x) node[fnode](n2){};
		\draw(1.75*\x,9*\x) node[fnode](n3){};
		\draw(-.5*\x,8*\x) node[fnode](n4){};
		\draw(1*\x,8*\x) node[fnode](n5){};
		\draw(.25*\x,7*\x) node[fnode](n6){};
		\draw(1.75*\x,7*\x) node[fnode](n7){};
		\draw(-.5*\x,6*\x) node[fnode](n8){};
		\draw(1*\x,6*\x) node[fnode](n9){};
		\draw(.25*\x,5*\x) node[fnode](n10){};
		\draw(1.75*\x,5*\x) node[fnode](n11){};
		\draw(1*\x,4*\x) node[fnode](n12){};
		\draw(2.5*\x,4*\x) node[fnode](n13){};
		\draw(1.75*\x,3*\x) node[fnode](n14){};
		\draw(2.5*\x,2*\x) node[fnode](n15){};
		\draw(3.25*\x,1*\x) node[fnode](n16){};
		%%
		\draw[edge](n1) -- (n2);
		\draw[edge](n1) -- (n3);
		\draw[edge](n2) -- (n4);
		\draw[edge](n2) -- (n5);
		\draw[edge](n5) -- (n6);
		\draw[edge](n5) -- (n7);
		\draw[edge](n6) -- (n8);
		\draw[edge](n7) -- (n9);
		\draw[edge](n8) -- (n10);
		\draw[edge](n9) -- (n11);
		\draw[edge](n10) -- (n12);
		\draw[edge](n11) -- (n13);
		\draw[edge](n13) -- (n14);
		\draw[edge](n14) -- (n15);
		\draw[edge](n15) -- (n16);
	\end{tikzpicture}
\end{page}

%Figure: Rotation of $\nu$-trees.
\begin{page}%page 6
	\begin{tikzpicture}
		\draw(1,1) node{\begin{tikzpicture}\small
			\def\x{.75};
			%%
			\draw[dashed,\gray](0*\x,0*\x) -- (1*\x,0*\x);
			\draw[dashed,\gray](0*\x,1*\x) -- (3*\x,1*\x);
			\draw[dashed,\gray](0*\x,2*\x) -- (3*\x,2*\x);
			\draw[dashed,\gray](0*\x,3*\x) -- (5*\x,3*\x);
			\draw[dashed,\gray](0*\x,4*\x) -- (5*\x,4*\x);
			\draw[dashed,\gray](0*\x,0*\x) -- (0*\x,4*\x);
			\draw[dashed,\gray](1*\x,0*\x) -- (1*\x,4*\x);
			\draw[dashed,\gray](2*\x,1*\x) -- (2*\x,4*\x);
			\draw[dashed,\gray](3*\x,1*\x) -- (3*\x,4*\x);
			\draw[dashed,\gray](4*\x,3*\x) -- (4*\x,4*\x);
			\draw[dashed,\gray](5*\x,3*\x) -- (5*\x,4*\x);
			%%
			\draw(1*\x,0*\x) node[fill,circle,scale=.6](v1){};
			\draw(3*\x,1*\x) node[fill,circle,scale=.6](v2){};
			\draw(1*\x,2*\x) node[fill=\blue,circle,scale=.6,label=below left:{\color{\blue}$q=p\llcorner r$}](v3){};
			\draw(2*\x,2*\x) node[fill,circle,scale=.6,label=below:$r$](v4){};
			\draw(3*\x,2*\x) node[fill,circle,scale=.6](v5){};
			\draw(4*\x,3*\x) node[fill,circle,scale=.6](v6){};
			\draw(5*\x,3*\x) node[fill,circle,scale=.6](v7){};
			\draw(0*\x,4*\x) node[fill,circle,scale=.6](v8){};
			\draw(1*\x,4*\x) node[fill,circle,scale=.6,label=above:$p$](v9){};
			\draw(4*\x,4*\x) node[fill,circle,scale=.6](v10){};
			%%
			\draw[line width=1pt,black](v1) -- (v3);
			\draw[line width=1pt,black](v2) -- (v5);
			\draw[line width=1pt,black](v3) -- (v4);
			\draw[line width=1pt,black](v3) -- (v9);
			\draw[line width=1pt,black](v4) -- (v5);
			\draw[line width=1pt,black](v6) -- (v7);
			\draw[line width=1pt,black](v6) -- (v10);
			\draw[line width=1pt,black](v8) -- (v9);
			\draw[line width=1pt,black](v9) -- (v10);
			%%
			\begin{pgfonlayer}{background}
				\fill[\gray,opacity=.5](1*\x,2*\x) -- (1*\x,4*\x) -- (2*\x,4*\x) -- (2*\x,2*\x) -- cycle;
			\end{pgfonlayer}
		\end{tikzpicture}};
		\draw(4,1) node{$\lessdot_{\nu}$};
		\draw(7,1) node{\begin{tikzpicture}\small
			\def\x{.75};
			%%
			\draw[dashed,\gray](0*\x,0*\x) -- (1*\x,0*\x);
			\draw[dashed,\gray](0*\x,1*\x) -- (3*\x,1*\x);
			\draw[dashed,\gray](0*\x,2*\x) -- (3*\x,2*\x);
			\draw[dashed,\gray](0*\x,3*\x) -- (5*\x,3*\x);
			\draw[dashed,\gray](0*\x,4*\x) -- (5*\x,4*\x);
			\draw[dashed,\gray](0*\x,0*\x) -- (0*\x,4*\x);
			\draw[dashed,\gray](1*\x,0*\x) -- (1*\x,4*\x);
			\draw[dashed,\gray](2*\x,1*\x) -- (2*\x,4*\x);
			\draw[dashed,\gray](3*\x,1*\x) -- (3*\x,4*\x);
			\draw[dashed,\gray](4*\x,3*\x) -- (4*\x,4*\x);
			\draw[dashed,\gray](5*\x,3*\x) -- (5*\x,4*\x);
			%%
			%%
			\draw(1*\x,0*\x) node[fill,circle,scale=.6](v1){};
			\draw(3*\x,1*\x) node[fill,circle,scale=.6](v2){};
			\draw(1*\x,2*\x) node[draw=\blue,circle,scale=.6]{};
			\draw(2*\x,2*\x) node[fill,circle,scale=.6,label=below:$r$](v3){};
			\draw(3*\x,2*\x) node[fill,circle,scale=.6](v4){};
			\draw(4*\x,3*\x) node[fill,circle,scale=.6](v5){};
			\draw(5*\x,3*\x) node[fill,circle,scale=.6](v6){};
			\draw(0*\x,4*\x) node[fill,circle,scale=.6](v7){};
			\draw(1*\x,4*\x) node[fill,circle,scale=.6,label=above:$p$](v8){};
			\draw(2*\x,4*\x) node[fill=\red,circle,scale=.6,label=above right:{\color{\red}$q'=p\urcorner r$}](v9){};
			\draw(4*\x,4*\x) node[fill,circle,scale=.6](v10){};
			%%
			\draw[line width=1pt,black](v1) -- (v8);
			\draw[line width=1pt,black](v2) -- (v4);
			\draw[line width=1pt,black](v3) -- (v4);
			\draw[line width=1pt,black](v3) -- (v9);
			\draw[line width=1pt,black](v5) -- (v6);
			\draw[line width=1pt,black](v5) -- (v10);
			\draw[line width=1pt,black](v7) -- (v8);
			\draw[line width=1pt,black](v8) -- (v9);
			\draw[line width=1pt,black](v9) -- (v10);
			%%
			\begin{pgfonlayer}{background}
				\fill[\gray,opacity=.5](1*\x,2*\x) -- (1*\x,4*\x) -- (2*\x,4*\x) -- (2*\x,2*\x) -- cycle;
			\end{pgfonlayer}
		\end{tikzpicture}};
	\end{tikzpicture}
\end{page}

%Figure: The right-flushing bijection.
\begin{page}%page 7
	\begin{tikzpicture}
		\def\tc{\redLabel};
		%%
		\draw(1,.9) node[pathnode]{\nuPath{1,3,2,5,3,1}{1,1,3,3,4,4,5,5,6,6}{}{.6}{}{}};
		\draw(-1.5,-1.1) node[\tc,scale=.75]{$p_{0}$};
		\draw(-1.5,-.5) node[\tc,scale=.75]{$p_{1}$};
		\draw(-.9,-.5) node[\tc,scale=.75]{$p_{2}$};
		\draw(-.3,-.5) node[\tc,scale=.75]{$p_{3}$};
		\draw(-.3,.1) node[\tc,scale=.75]{$p_{4}$};
		\draw(-.3,.7) node[\tc,scale=.75]{$p_{5}$};
		\draw(.3,.7) node[\tc,scale=.75]{$p_{6}$};
		\draw(.9,.7) node[\tc,scale=.75]{$p_{7}$};
		\draw(.9,1.3) node[\tc,scale=.75]{$p_{8}$};
		\draw(1.5,1.3) node[\tc,scale=.75]{$p_{9}$};
		\draw(2.15,1.3) node[\tc,scale=.75]{$p_{10}$};
		\draw(2.15,1.9) node[\tc,scale=.75]{$p_{11}$};
		\draw(2.75,1.9) node[\tc,scale=.75]{$p_{12}$};
		\draw(3.35,1.9) node[\tc,scale=.75]{$p_{13}$};
		\draw(3.35,2.5) node[\tc,scale=.75]{$p_{14}$};
		\draw(3.95,2.5) node[\tc,scale=.75]{$p_{15}$};
		%%
		\draw(5.25,1) node{$\overset{\text{right-flushing}}{\longrightarrow}$};
		%%
		\draw(10,1) node[pathnode]{\nuTree{1,3,2,5,3,1}{0/0//,1/1//,2/1//d,3/1//d,1/2//,6/3//,7/3//d,8/3//d,4/4//,5/4//d,6/4//d,0/5//,1/5//d,4/5//d,0/6//,9/6//d}{p1/p12,p2/p3,p2/p5,p3/p4,p5/p13,p6/p7,p6/p11,p7/p8,p9/p10,p9/p14,p10/p11,p12/p15,p12/p13,p13/p14,p15/p16}{}{.6}{}{}};
		\draw(7.5,-1.15) node[\tc,scale=.75]{$\overline{p}_{0}$};
		\draw(8.1,-.55) node[\tc,scale=.75]{$\overline{p}_{3}$};
		\draw(8.7,-.55) node[\tc,scale=.75]{$\overline{p}_{2}$};
		\draw(9.3,-.55) node[\tc,scale=.75]{$\overline{p}_{1}$};
		\draw(8.1,.05) node[\tc,scale=.75]{$\overline{p}_{4}$};
		\draw(11.1,.65) node[\tc,scale=.75]{$\overline{p}_{7}$};
		\draw(11.7,.65) node[\tc,scale=.75]{$\overline{p}_{6}$};
		\draw(12.3,.65) node[\tc,scale=.75]{$\overline{p}_{5}$};
		\draw(9.95,1.25) node[\tc,scale=.75]{$\overline{p}_{10}$};
		\draw(10.5,1.25) node[\tc,scale=.75]{$\overline{p}_{9}$};
		\draw(11.1,1.25) node[\tc,scale=.75]{$\overline{p}_{8}$};
		\draw(7.55,1.85) node[\tc,scale=.75]{$\overline{p}_{13}$};
		\draw(8.15,1.85) node[\tc,scale=.75]{$\overline{p}_{12}$};
		\draw(9.95,1.85) node[\tc,scale=.75]{$\overline{p}_{11}$};
		\draw(7.55,2.45) node[\tc,scale=.75]{$\overline{p}_{15}$};
		\draw(12.95,2.45) node[\tc,scale=.75]{$\overline{p}_{14}$};
	\end{tikzpicture}
\end{page}

%Figure: The $\nu$-associahedron for $\nu$=EENEN.
\begin{page}%page 8
	\begin{tikzpicture}[scale=.7]\small
		\def\x{1.8};
		\def\sb{.4};
		\def\sm{.25};
		\def\bc{\gray!50!white};
		%%
		\draw(5*\x,1*\x) node[pathnode](n1){\nuTree{0,2,1,1,1}{0/0//,1/0//,2/0//,0/1//,3/1//,0/2//}{p1/p2,p1/p4,p2/p3,p4/p5,p4/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{y^{2}}$/}{}};
		\draw(8*\x,4*\x) node[pathnode](n2){\nuTree{0,2,1,1,1}{1/0//,2/0//,0/1//,1/1//,3/1//,0/2//}{p1/p2,p1/p4,p3/p4,p4/p5,p3/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{xy}$/}{}};
		\draw(1*\x,5*\x) node[pathnode](n3){\nuTree{0,2,1,1,1}{0/0//,1/0//,2/0///,3/1//,0/2//,3/2//}{p1/p2,p1/p5,p2/p3,p4/p6,p5/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{xy}$/}{}};
		\draw(8*\x,6*\x) node[pathnode](n4){\nuTree{0,2,1,1,1}{1/0//,2/0//,1/1//,3/1//,0/2//,1/2//}{p1/p2,p1/p3,p3/p4,p3/p6,p5/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{x^{2}}$/}{}};
		\draw(10*\x,6*\x) node[pathnode](n5){\nuTree{0,2,1,1,1}{2/0//,0/1//,1/1//,2/1//,3/1//,0/2//}{p1/p4,p2/p3,p2/p6,p3/p4,p4/p5}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{xy}$/}{}};
		\draw(10*\x,8*\x) node[pathnode](n6){\nuTree{0,2,1,1,1}{2/0//,1/1//,2/1//,3/1//,0/2//,1/2//}{p1/p3,p2/p3,p2/p6,p3/p4,p5/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{x^{2}}$/}{}};
		\draw(5*\x,9*\x) node[pathnode](n7){\nuTree{0,2,1,1,1}{1/0//,2/0//,3/1//,0/2//,1/2//,3/2//}{p1/p2,p1/p5,p3/p6,p4/p5,p5/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{x^{2}}$/}{}};
		\draw(10*\x,10*\x) node[pathnode](n8){\nuTree{0,2,1,1,1}{2/0//,2/1//,3/1//,0/2//,1/2//,2/2//}{p1/p2,p2/p3,p2/p6,p4/p5,p5/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{x^{2}}$/}{}};
		\draw(8*\x,12*\x) node[pathnode](n9){\nuTree{0,2,1,1,1}{2/0//,3/1//,0/2//,1/2//,2/2//,3/2//}{p1/p5,p2/p6,p3/p4,p4/p5,p5/p6}{0/0/\red,0/1/\red}{\sb}{2.25/.25/$\mathbf{x^{2}}$/}{}};
		%%
		\draw(n1) -- (n2) node[pathnode,fill=\bc] at (6.5*\x,2.5*\x) 
			{\nuTree{0,2,1,1,1}{1/0/,2/0/,0/1/,3/1/,0/2/}{p1/p2,p3/p4,p3/p5}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{y}$/}{}};
		\draw(n1) -- (n3) node[pathnode,fill=\bc] at (3*\x,3*\x)
			{\nuTree{0,2,1,1,1}{0/0/,1/0/,2/0/,3/1/,0/2/}{p1/p2,p1/p5,p2/p3}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{y}$/}{}};
		\draw(n2) -- (n4) node[pathnode,fill=\bc] at (8*\x,5*\x) 
			{\nuTree{0,2,1,1,1}{1/0/,2/0/,1/1/,3/1/,0/2/}{p1/p2,p1/p3,p3/p4}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		\draw(n2) -- (n5) node[pathnode,fill=\bc] at (9*\x,5*\x)
			{\nuTree{0,2,1,1,1}{2/0/,0/1/,1/1/,3/1/,0/2/}{p2/p3,p2/p5,p3/p4}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{y}$/}{}};
		\draw(n3) -- (n7) node[pathnode,fill=\bc] at (3*\x,7*\x)
			{\nuTree{0,2,1,1,1}{1/0/,2/0/,3/1/,0/2/,3/2/}{p1/p2,p3/p5,p4/p5}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		\draw(n4) -- (n6) node[pathnode,fill=\bc] at (9*\x,7*\x)
			{\nuTree{0,2,1,1,1}{2/0/,1/1/,3/1/,0/2/,1/2/}{p2/p3,p2/p5,p4/p5}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		\draw(n4) -- (n7) node[pathnode,fill=\bc] at (6.5*\x,7.5*\x)
			{\nuTree{0,2,1,1,1}{1/0/,2/0/,3/1/,0/2/,1/2/}{p1/p2,p1/p5,p4/p5}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		\draw(n5) -- (n6) node[pathnode,fill=\bc] at (10*\x,7*\x)
			{\nuTree{0,2,1,1,1}{2/0/,1/1/,2/1/,3/1/,0/2/}{p1/p3,p2/p3,p3/p4}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		\draw(n6) -- (n8) node[pathnode,fill=\bc] at (10*\x,9*\x)
			{\nuTree{0,2,1,1,1}{2/0/,2/1/,3/1/,0/2/,1/2/}{p1/p2,p2/p3,p4/p5}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		\draw(n7) -- (n9) node[pathnode,fill=\bc] at (6.5*\x,10.5*\x)
			{\nuTree{0,2,1,1,1}{2/0/,3/1/,0/2/,1/2/,3/2/}{p2/p5,p3/p4,p4/p5}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		\draw(n8) -- (n9) node[pathnode,fill=\bc] at (9*\x,11*\x)
			{\nuTree{0,2,1,1,1}{2/0/,3/1/,0/2/,1/2/,2/2/}{p1/p5,p3/p4,p4/p5}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{x}$/}{}};
		%%
		\draw(5*\x,5*\x) node[pathnode]{\nuTree{0,2,1,1,1}{1/0/,2/0/,3/1/,0/2/}{p1/p2}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{1}$/}{}};
		\draw(9*\x,6*\x) node[pathnode]{\nuTree{0,2,1,1,1}{2/0/,1/1/,3/1/,0/2/}{p2/p3}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{1}$/}{}};
		\draw(8*\x,9*\x) node[pathnode]{\nuTree{0,2,1,1,1}{2/0/,3/1/,0/2/,1/2/}{p3/p4}{0/0/\red,0/1/\red}{\sm}{2.25/.25/$\mathbf{1}$/}{}};
		%%
		\begin{pgfonlayer}{background}
			\fill[\bc,rounded corners](5.5*\x,.5*\x) -- (4.5*\x,.5*\x) -- (1.2*\x,3.75*\x) -- (1.2*\x,3.9*\x) -- (4.95*\x,7.6*\x) -- (5.05*\x,7.6*\x) -- (7.3*\x,5.4*\x) -- (7.3*\x,2.3*\x) -- cycle;
			\fill[\bc,rounded corners](7.4*\x,2.4*\x) -- (7.4*\x,5.4*\x) -- (8.65*\x,5.4*\x) -- (8.65*\x,6.5*\x) -- (9.4*\x,6.5*\x) -- (9.4*\x,4.5*\x) -- cycle;
			\fill[\bc,rounded corners](9.45*\x,4.55*\x) -- (9.45*\x,7.4*\x) -- (10.55*\x,7.4*\x) -- (10.55*\x,5.7*\x) -- cycle;
			\fill[\bc,rounded corners](9.45*\x,7.5*\x) -- (9.45*\x,9.4*\x) -- (10.55*\x,9.4*\x) -- (10.55*\x,7.5*\x) -- cycle;
			\fill[\bc,rounded corners](7.3*\x,5.5*\x) -- (5.05*\x,7.7*\x) -- (8.05*\x,10.7*\x) -- (8.15*\x,10.7*\x) -- (9.4*\x,9.45*\x) -- (9.4*\x,6.6*\x) -- (8.6*\x,6.6*\x) -- (8.6*\x,5.5*\x) -- cycle;
			\fill[\bc,rounded corners](.4*\x,4.5*\x) -- (.4*\x,5.5*\x) -- (3.8*\x,8.8*\x) -- (4.95*\x,7.7*\x) -- (1.1*\x,3.9*\x) -- cycle;
			\fill[\bc,rounded corners](3.85*\x,8.85*\x) -- (6.9*\x,11.95*\x) -- (8.05*\x,10.8*\x) -- (5.05*\x,7.8*\x) -- (4.95*\x,7.8*\x) -- cycle;
			\fill[\bc,rounded corners](9.45*\x,9.5*\x) -- (8.15*\x,10.8*\x) -- (9.15*\x,11.8*\x) -- (9.5*\x,11.5*\x) -- (10.55*\x,10.5*\x) -- (10.55*\x,9.5*\x) -- cycle;
			\fill[\bc,rounded corners](6.95*\x,12*\x) -- (7.45*\x,12.5*\x) -- (8.45*\x,12.5*\x) -- (9.1*\x,11.85*\x) -- (8.15*\x,10.9*\x) -- (8.1*\x,10.85*\x) -- cycle;
		\end{pgfonlayer}
	\end{tikzpicture}
\end{page}

%Figure: Another $\nu-$associahedron.
\begin{page}%page 9
	\begin{tikzpicture}
		\def\x{1};
		\def\y{1};
		\def\ps{.5};
		%%
		\coordinate (c1030) at (-21*\x,0*\y);
			\node[pathnode] (n1030) at (c1030)  {\nuPath{0,1,1,2,1,1,1}{0,2,2,2,3}{1/0/\redLabel/\blueLabel,4/2/\redLabel/\blueLabel}{\ps}{1.35/0/$x^{2}y^{2}$/\blueLabel,1.35/.65/$x(y{+}1)^{2}$/\redLabel}{\gray!50!white}};
		\coordinate (c1120) at (-10.5*\x,0*\y);
			\node[pathnode,draw,circle] (n1120) at (c1120)  {\nuPath{0,1,1,2,1,1,1}{0,1,2,2,3}{1/0/\redLabel/\blueLabel,2/1//\blueLabel,4/2/\redLabel/\blueLabel}{\ps}{1.35/0/$x^{3}y^{2}$/\blueLabel,1.35/.65/$(x{+}1)(y{+}1)^{2}$/\redLabel}{\gray!50!white}};
		\coordinate (c1210) at (0*\x,0*\y);
			\node[pathnode] (n1210) at (c1210)  {\nuPath{0,1,1,2,1,1,1}{0,1,1,2,3}{1/0/\redLabel/\blueLabel,3/1/\redLabel/\blueLabel,4/2/\redLabel/\blueLabel}{\ps}{1.35/0/$x^{3}y^{3}$/\blueLabel,1.35/.65/$(y{+}1)^{3}$/\redLabel}{\gray!50!white}};
		\coordinate (c1021) at (-24*\x,4*\y);
			\node[pathnode] (n1021) at (c1021)  {\nuPath{0,1,1,2,1,1,1}{0,2,2,3,3}{1/0/\redLabel/\blueLabel,3/2//\blueLabel}{\ps}{1.35/0/$x^{2}y$/\blueLabel,1.35/.65/$x(x{+}1)(y{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c1111) at (-13.5*\x,4*\y);
			\begin{pgfonlayer}{background}
				\node[pathnode] (n1111) at (c1111)  {\nuPath{0,1,1,2,1,1,1}{0,1,2,3,3}{1/0/\redLabel/\blueLabel,2/1//\blueLabel,3/2//\blueLabel}{\ps}{1.35/0/$x^{3}y$/\blueLabel,1.35/.65/$(x{+}1)^{2}(y{+}1)$/\redLabel}{\gray!50!white}};
			\end{pgfonlayer}
		\coordinate (c0310) at (0*\x,5*\y);
			\node[pathnode,draw,circle] (n0310) at (c0310)  {\nuPath{0,1,1,2,1,1,1}{1,1,1,2,3}{3/1/\redLabel/\blueLabel,4/2/\redLabel/\blueLabel}{\ps}{1.35/0/$x^{2}y^{2}$/\blueLabel,1.35/.65/$x(y{+}1)^{2}$/\redLabel}{\gray!50!white}};
		\coordinate (c1012) at (-25.5*\x,6*\y);
			\node[pathnode] (n1012) at (c1012)  {\nuPath{0,1,1,2,1,1,1}{0,2,3,3,3}{1/0/\redLabel/\blueLabel,2/2//\blueLabel}{\ps}{1.35/0/$x^{2}y$/\blueLabel,1.35/.65/$x(x{+}1)(y{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c1003) at (-23.5*\x,8*\y);
			\begin{pgfonlayer}{background}
				\node[pathnode] (n1003) at (c1003)  {\nuPath{0,1,1,2,1,1,1}{0,3,3,3,3}{1/0/\redLabel/\blueLabel}{\ps}{1.35/0/$xy$/\blueLabel,1.35/.65/$x^{2}(y{+}1)$/\redLabel}{\gray!50!white}};
			\end{pgfonlayer}
		\coordinate (c1102) at (-9.5*\x,8*\y);
			\begin{pgfonlayer}{background}
				\node[pathnode] (n1102) at (c1102)  {\nuPath{0,1,1,2,1,1,1}{0,1,3,3,3}{1/0/\redLabel/\blueLabel,2/1//\blueLabel}{\ps}{1.35/0/$x^{2}y$/\blueLabel,1.35/.65/$x(x{+}1)(y{+}1)$/\redLabel}{\gray!50!white}};
			\end{pgfonlayer}
		\coordinate (c1201) at (8*\x,8*\y);
			\node[pathnode] (n1201) at (c1201)  {\nuPath{0,1,1,2,1,1,1}{0,1,1,3,3}{1/0/\redLabel/\blueLabel,3/1/\redLabel/\blueLabel}{\ps}{1.35/0/$x^{2}y^{2}$/\blueLabel,1.35/.65/$x(y{+}1)^{2}$/\redLabel}{\gray!50!white}};
		\coordinate (c0040) at (-21*\x,10*\y);
			\node[pathnode] (n0040) at (c0040)  {\nuPath{0,1,1,2,1,1,1}{2,2,2,2,3}{4/2/\redLabel/\blueLabel}{\ps}{1.35/0/$xy$/\blueLabel,1.35/.65/$x^{2}(y{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0130) at (-10.5*\x,10*\y);
			\node[pathnode] (n0130) at (c0130)  {\nuPath{0,1,1,2,1,1,1}{1,2,2,2,3}{1/1//\blueLabel,4/2/\redLabel/\blueLabel}{\ps}{1.35/0/$x^{2}y$/\blueLabel,1.35/.65/$x(x{+}1)(y{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0220) at (-3.5*\x,10*\y);
			\node[pathnode] (n0220) at (c0220)  {\nuPath{0,1,1,2,1,1,1}{1,1,2,2,3}{2/1//\blueLabel,4/2/\redLabel/\blueLabel}{\ps}{1.35/0/$x^{2}y$/\blueLabel,1.35/.65/$x(x{+}1)(y{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0301) at (8*\x,13*\y);
			\node[pathnode] (n0301) at (c0301)  {\nuPath{0,1,1,2,1,1,1}{1,1,1,3,3}{3/1/\redLabel/\blueLabel}{\ps}{1.35/0/$xy$/\blueLabel,1.35/.65/$x^{2}(y{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0031) at (-22.5*\x,17*\y);
			\node[pathnode] (n0031) at (c0031)  {\nuPath{0,1,1,2,1,1,1}{2,2,2,3,3}{3/2//\blueLabel}{\ps}{1.35/0/$x$/\blueLabel,1.35/.65/$x^{2}(x{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0121) at (-12*\x,17*\y);
			\node[pathnode] (n0121) at (c0121)  {\nuPath{0,1,1,2,1,1,1}{1,2,2,3,3}{1/1//\blueLabel,3/2//\blueLabel}{\ps}{1.35/0/$x^{2}$/\blueLabel,1.35/.65/$x(x{+}1)^{2}$/\redLabel}{\gray!50!white}};
		\coordinate (c0211) at (-5*\x,17*\y);
			\node[pathnode] (n0211) at (c0211)  {\nuPath{0,1,1,2,1,1,1}{1,1,2,3,3}{2/1//\blueLabel,3/2//\blueLabel}{\ps}{1.35/0/$x^{2}$/\blueLabel,1.35/.65/$x(x{+}1)^{2}$/\redLabel}{\gray!50!white}};
		\coordinate (c0022) at (-24*\x,19*\y);
			\node[pathnode] (n0022) at (c0022)  {\nuPath{0,1,1,2,1,1,1}{2,2,3,3,3}{2/2//\blueLabel}{\ps}{1.35/0/$x$/\blueLabel,1.35/.65/$x^{2}(x{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0112) at (-13.5*\x,19*\y);
			\node[pathnode] (n0112) at (c0112)  {\nuPath{0,1,1,2,1,1,1}{1,2,3,3,3}{1/1//\blueLabel,2/2//\blueLabel}{\ps}{1.35/0/$x^{2}$/\blueLabel,1.35/.65/$x(x{+}1)^{2}$/\redLabel}{\gray!50!white}};
		\coordinate (c0013) at (-25.5*\x,21*\y);
			\node[pathnode] (n0013) at (c0013)  {\nuPath{0,1,1,2,1,1,1}{2,3,3,3,3}{1/2//\blueLabel}{\ps}{1.35/0/$x$/\blueLabel,1.35/.65/$x^{2}(x{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0004) at (-23.5*\x,23*\y);
			\node[pathnode] (n0004) at (c0004)  {\nuPath{0,1,1,2,1,1,1}{3,3,3,3,3}{}{\ps}{1.35/0/$1$/\blueLabel,1.35/.65/$x^{3}$/\redLabel}{\gray!50!white}};
		\coordinate (c0103) at (-9.5*\x,23*\y);
			\node[pathnode] (n0103) at (c0103)  {\nuPath{0,1,1,2,1,1,1}{1,3,3,3,3}{1/1//\blueLabel}{\ps}{1.35/0/$x$/\blueLabel,1.35/.65/$x^{2}(x{+}1)$/\redLabel}{\gray!50!white}};
		\coordinate (c0202) at (1*\x,23*\y);
			\node[pathnode] (n0202) at (c0202)  {\nuPath{0,1,1,2,1,1,1}{1,1,3,3,3}{2/1//\blueLabel}{\ps}{1.35/0/$x$/\blueLabel,1.35/.65/$x^{2}(x{+}1)$/\redLabel}{\gray!50!white}};
		%%
		%% Drawing edges in the back
		%% List of edges in the back
		\def\listEdgesBack{n1003/n0004,n1102/n1003,n1012/n1003,n1201/n1102,n1111/n1102,n1102/n0103,n1120/n1111,n1111/n1021,n1111/n0112};
		\foreach \a/\b in \listEdgesBack{
			\begin{pgfonlayer}{background}
				\draw[edge,back,->] (\a) -- (\b);
			\end{pgfonlayer}
		}
		%% Drawing edges in the front
		%% List of edges in the front
		\def\listEdgesFront{n1210/n1201,n1210/n1120,n1210/n0310,n1201/n0301,n1120/n1030,n1120/n0130,n1030/n1021,n1030/n0040,n1021/n1012,n1021/n0022,n1012/n0013,n0310/n0301,n0310/n0220,n0301/n0202,n0220/n0211,n0220/n0130,n0211/n0202,n0211/n0121,n0202/n0103,n0130/n0121,n0130/n0040,n0121/n0112,n0121/n0031,n0112/n0103,n0112/n0022,n0103/n0004,n0040/n0031,n0031/n0022,n0022/n0013,n0013/n0004};
		%%
		%%
		\foreach \a/\b in \listEdgesFront{
			\draw[edge,->] (\a) -- (\b);
		}

	\end{tikzpicture}
\end{page}

%Illustration of Proposition 4.1
\begin{page}%page 10
	\begin{tikzpicture}
		\def\x{1};
		\def\y{1};
		\def\s{.75};		%scales the nodes
		\def\ps{.5};		%scales the paths/trees
		\def\ls{1.8};		%scales the labels
		\def\drx{0};		%shift x right
		\def\dry{-27};		%shift y right
		\def\dlx{0};		%shift x left
		\def\dly{22};		%shift y left
		%%
		\coordinate (c1030) at (-21*\x,0*\y);
			\node[pathnode] (n1030) at (c1030)  {\nuPath{0,1,1,2,1,1,1}{0,2,2,2,3}{1/0/\redLabel/\blue,4/2/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-22*\x,.35*\y) {$\mathbf{x(y{+}1)^{2}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-22*\x,-.35*\y) {$\mathbf{x^{2}y^{2}}$};
		\coordinate (c1120) at (-10.5*\x,0*\y);
			\node[pathnode,draw,circle] (n1120) at (c1120)  {\nuPath{0,1,1,2,1,1,1}{0,1,2,2,3}{1/0/\redLabel/\blue,2/1//\blue,4/2/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-11.5*\x,.35*\y) {$\mathbf{(x{+}1)(y{+}1)^{2}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-11.5*\x,-.35*\y) {$\mathbf{x^{3}y^{2}}$};
		\coordinate (c1210) at (0*\x,0*\y);
			\node[pathnode] (n1210) at (c1210)  {\nuPath{0,1,1,2,1,1,1}{0,1,1,2,3}{1/0/\redLabel/\blue,3/1/\redLabel/\blue,4/2/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-1*\x,.35*\y) {$\mathbf{(y{+}1)^{3}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-1*\x,-.35*\y) {$\mathbf{x^{3}y^{3}}$};
		\coordinate (c1021) at (-24*\x,4*\y);
			\node[pathnode] (n1021) at (c1021)  {\nuPath{0,1,1,2,1,1,1}{0,2,2,3,3}{1/0/\redLabel/\blue,3/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-25*\x,4.35*\y) {$\mathbf{x(x{+}1)(y{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-25*\x,3.65*\y) {$\mathbf{x^{2}y}$};
		\coordinate (c1111) at (-13.5*\x,4*\y);
			\begin{pgfonlayer}{background}
				\node[pathnode] (n1111) at (c1111)  {\nuPath{0,1,1,2,1,1,1}{0,1,2,3,3}{1/0/\redLabel/\blue,2/1//\blue,3/2//\blue}{\ps}{}{\gray!50!white}};
				\node[anchor=east,\redLabel,scale=\ls] at (-14.5*\x,4.35*\y) {$\mathbf{(x{+}1)^{2}(y{+}1)}$};
				\node[anchor=east,\blueLabel,scale=\ls] at (-14.5*\x,3.65*\y) {$\mathbf{x^{3}y}$};
			\end{pgfonlayer}
		\coordinate (c0310) at (0*\x,5*\y);
			\node[pathnode,draw,circle] (n0310) at (c0310)  {\nuPath{0,1,1,2,1,1,1}{1,1,1,2,3}{3/1/\redLabel/\blue,4/2/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-1*\x,5.35*\y) {$\mathbf{x(y{+}1)^{2}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-1*\x,4.65*\y) {$\mathbf{x^{2}y^{2}}$};
		\coordinate (c1012) at (-25.5*\x,6*\y);
			\node[pathnode] (n1012) at (c1012)  {\nuPath{0,1,1,2,1,1,1}{0,2,3,3,3}{1/0/\redLabel/\blue,2/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-26.5*\x,6.35*\y) {$\mathbf{x(x{+}1)(y{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-26.5*\x,5.65*\y) {$\mathbf{x^{2}y}$};
		\coordinate (c1003) at (-23.5*\x,8*\y);
			\begin{pgfonlayer}{background}
				\node[pathnode] (n1003) at (c1003)  {\nuPath{0,1,1,2,1,1,1}{0,3,3,3,3}{1/0/\redLabel/\blue}{\ps}{}{\gray!50!white}};
				\node[anchor=east,\redLabel,scale=\ls] at (-24.5*\x,8.35*\y) {$\mathbf{x^{2}(y{+}1)}$};
				\node[anchor=east,\blueLabel,scale=\ls] at (-24.5*\x,7.65*\y) {$\mathbf{xy}$};
			\end{pgfonlayer}
		\coordinate (c1102) at (-9.5*\x,8*\y);
			\begin{pgfonlayer}{background}
				\node[pathnode] (n1102) at (c1102)  {\nuPath{0,1,1,2,1,1,1}{0,1,3,3,3}{1/0/\redLabel/\blue,2/1//\blue}{\ps}{}{\gray!50!white}};
				\node[anchor=east,\redLabel,scale=\ls] at (-10.5*\x,8.35*\y) {$\mathbf{x(x{+}1)(y{+}1)}$};
				\node[anchor=east,\blueLabel,scale=\ls] at (-10.5*\x,7.65*\y) {$\mathbf{x^{2}y}$};
			\end{pgfonlayer}
		\coordinate (c1201) at (8*\x,8*\y);
			\node[pathnode] (n1201) at (c1201)  {\nuPath{0,1,1,2,1,1,1}{0,1,1,3,3}{1/0/\redLabel/\blue,3/1/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (7*\x,8.35*\y) {$\mathbf{x(y{+}1)^{2}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (7*\x,7.65*\y) {$\mathbf{x^{2}y^{2}}$};
		\coordinate (c0040) at (-21*\x,10*\y);
			\node[pathnode] (n0040) at (c0040)  {\nuPath{0,1,1,2,1,1,1}{2,2,2,2,3}{4/2/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-22*\x,10.35*\y) {$\mathbf{x^{2}(y{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-22*\x,9.65*\y) {$\mathbf{xy}$};
		\coordinate (c0130) at (-10.5*\x,10*\y);
			\node[pathnode] (n0130) at (c0130)  {\nuPath{0,1,1,2,1,1,1}{1,2,2,2,3}{1/1//\blue,4/2/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-11.5*\x,10.35*\y) {$\mathbf{x(x{+}1)(y{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-11.5*\x,9.65*\y) {$\mathbf{x^{2}y}$};
		\coordinate (c0220) at (-3.5*\x,10*\y);
			\node[pathnode] (n0220) at (c0220)  {\nuPath{0,1,1,2,1,1,1}{1,1,2,2,3}{2/1//\blue,4/2/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-4.5*\x,10.35*\y) {$\mathbf{x(x{+}1)(y{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-4.5*\x,9.65*\y) {$\mathbf{x^{2}y}$};
		\coordinate (c0301) at (8*\x,13*\y);
			\node[pathnode] (n0301) at (c0301)  {\nuPath{0,1,1,2,1,1,1}{1,1,1,3,3}{3/1/\redLabel/\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (7*\x,13.35*\y) {$\mathbf{x^{2}(y{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (7*\x,12.65*\y) {$\mathbf{xy}$};
		\coordinate (c0031) at (-22.5*\x,17*\y);
			\node[pathnode] (n0031) at (c0031)  {\nuPath{0,1,1,2,1,1,1}{2,2,2,3,3}{3/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-23.5*\x,17.35*\y) {$\mathbf{x^{2}(x{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-23.5*\x,16.65*\y) {$\mathbf{x}$};
		\coordinate (c0121) at (-12*\x,17*\y);
			\node[pathnode] (n0121) at (c0121)  {\nuPath{0,1,1,2,1,1,1}{1,2,2,3,3}{1/1//\blue,3/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-13*\x,17.35*\y) {$\mathbf{x(x{+}1)^{2}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-13*\x,16.65*\y) {$\mathbf{x^{2}}$};
		\coordinate (c0211) at (-5*\x,17*\y);
			\node[pathnode] (n0211) at (c0211)  {\nuPath{0,1,1,2,1,1,1}{1,1,2,3,3}{2/1//\blue,3/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-6*\x,17.35*\y) {$\mathbf{x(x{+}1)^{2}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-6*\x,16.65*\y) {$\mathbf{x^{2}}$};
		\coordinate (c0022) at (-24*\x,19*\y);
			\node[pathnode] (n0022) at (c0022)  {\nuPath{0,1,1,2,1,1,1}{2,2,3,3,3}{2/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-25*\x,19.35*\y) {$\mathbf{x^{2}(x{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-25*\x,18.65*\y) {$\mathbf{x}$};
		\coordinate (c0112) at (-13.5*\x,19*\y);
			\node[pathnode] (n0112) at (c0112)  {\nuPath{0,1,1,2,1,1,1}{1,2,3,3,3}{1/1//\blue,2/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-14.5*\x,19.35*\y) {$\mathbf{x(x{+}1)^{2}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-14.5*\x,18.65*\y) {$\mathbf{x^{2}}$};
		\coordinate (c0013) at (-25.5*\x,21*\y);
			\node[pathnode] (n0013) at (c0013)  {\nuPath{0,1,1,2,1,1,1}{2,3,3,3,3}{1/2//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-26.5*\x,21.35*\y) {$\mathbf{x^{2}(x{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-26.5*\x,20.65*\y) {$\mathbf{x}$};
		\coordinate (c0004) at (-23.5*\x,23*\y);
			\node[pathnode] (n0004) at (c0004)  {\nuPath{0,1,1,2,1,1,1}{3,3,3,3,3}{}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-24.5*\x,23.35*\y) {$\mathbf{x^{3}}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-24.5*\x,22.65*\y) {$\mathbf{1}$};
		\coordinate (c0103) at (-9.5*\x,23*\y);
			\node[pathnode] (n0103) at (c0103)  {\nuPath{0,1,1,2,1,1,1}{1,3,3,3,3}{1/1//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (-10.5*\x,23.35*\y) {$\mathbf{x^{2}(x{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (-10.5*\x,22.65*\y) {$\mathbf{x}$};
		\coordinate (c0202) at (1*\x,23*\y);
			\node[pathnode] (n0202) at (c0202)  {\nuPath{0,1,1,2,1,1,1}{1,1,3,3,3}{2/1//\blue}{\ps}{}{\gray!50!white}};
			\node[anchor=east,\redLabel,scale=\ls] at (0*\x,23.35*\y) {$\mathbf{x^{2}(x{+}1)}$};
			\node[anchor=east,\blueLabel,scale=\ls] at (0*\x,22.65*\y) {$\mathbf{x}$};
		%%
		%% Drawing edges in the back
		%% List of edges in the back
		\def\listEdgesBack{n1003/n0004,n1102/n1003,n1012/n1003,n1201/n1102,n1111/n1102,n1102/n0103,n1120/n1111,n1111/n1021,n1111/n0112};
		\foreach \a/\b in \listEdgesBack{
			\begin{pgfonlayer}{background}
				\draw[edge,back,->] (\a) -- (\b);
			\end{pgfonlayer}
		}
		%% Drawing edges in the front
		%% List of edges in the front
		\def\listEdgesFront{n1210/n1201,n1210/n1120,n1210/n0310,n1201/n0301,n1120/n1030,n1120/n0130,n1030/n1021,n1030/n0040,n1021/n1012,n1021/n0022,n1012/n0013,n0310/n0301,n0310/n0220,n0301/n0202,n0220/n0211,n0220/n0130,n0211/n0202,n0211/n0121,n0202/n0103,n0130/n0121,n0130/n0040,n0121/n0112,n0121/n0031,n0112/n0103,n0112/n0022,n0103/n0004,n0040/n0031,n0031/n0022,n0022/n0013,n0013/n0004};
		%%
		%%
		\foreach \a/\b in \listEdgesFront{
			\draw[edge,->] (\a) -- (\b);
		}
		%% Highlighting some facets
		\begin{pgfonlayer}{middle}
% 			\fill[facetR](c0310) -- (c0301) -- (c0202) -- (c0103) -- (c0112) -- (c0121) -- (c0211) -- (c0220) -- cycle;
			\fill[facet](c0310) -- (c0301) -- (c0202) -- (c0211) -- (c0220) -- cycle;
			\fill[facet](c1030) -- (c1120) -- (c0130) -- (c0121) -- (c0112) -- (c0022) -- (c1021) -- cycle;
		\end{pgfonlayer}
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%%shift facets to the right%%
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%%
		\coordinate (rc0310) at ({(0+\drx)*\x},{(5+\dry)*\y});
			\node[pathnode,draw,circle] (rn0310) at (rc0310)  {\nuTree{0,1,1,2,1,1,1}{1/0/black,0/1/\blue,1/1/black,2/1/black,3/1/black,0/2/\blue,4/2/black,0/3/black}{p1/p3,p2/p3,p3/p4,p4/p5,p2/p6,p6/p7,p6/p8}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(0+\drx)*\x},{(4.4+\dry)*\y}) {$\mathbf{xy^{2}}$};
		\coordinate (rc0220) at ({(-3.5+\drx)*\x},{(10+\dry)*\y});
			\node[fnode,\green] (rn0220) at (rc0220)  {};
		\coordinate (rc0301) at ({(8+\drx)*\x},{(13+\dry)*\y});
			\node[fnode,\green] (rn0301) at (rc0301)  {};
% 		\coordinate (rc0121) at ({(-12+\drx)*\x},17*\y);
% 			\node[fnode,\green] (rn0121) at (rc0121)  {};
		\coordinate (rc0211) at ({(-5+\drx)*\x},{(17+\dry)*\y});
			\node[fnode,\green] (rn0211) at (rc0211)  {};
% 		\coordinate (rc0112) at ({(-13.5+\drx)*\x},19*\y);
% 			\node[fnode,\green] (rn0112) at (rc0112)  {};
% 		\coordinate (rc0103) at ({(-9.5+\drx)*\x},23*\y);
% 			\node[fnode,\green] (rn0103) at (rc0103)  {};
		\coordinate (rc0202) at ({(1+\drx)*\x},{(23+\dry)*\y});
			\node[fnode,\green] (rn0202) at (rc0202)  {};
		%%
		\draw[edge,\green,->](rn0310) -- (rn0301);
		\draw[edge,\green,->](rn0310) -- (rn0220);
		\draw[edge,\green,->](rn0220) -- (rn0211);
% 		\draw[edge,\green,->](rn0211) -- (rn0121);
		\draw[edge,\green,->](rn0211) -- (rn0202);
% 		\draw[edge,\green,->](rn0121) -- (rn0112);
% 		\draw[edge,\green,->](rn0112) -- (rn0103);
% 		\draw[edge,\green,->](rn0202) -- (rn0103);
		\draw[edge,\green,->](rn0301) -- (rn0202);
		%%
		\node[pathnode] at ({(-1.75+\drx)*\x},{(7.5+\dry)*\y})  {\nuTree{0,1,1,2,1,1,1}{1/0/black,1/1/black,2/1/black,3/1/black,0/2/\blue,4/2/black,0/3/black}{p1/p2,p2/p3,p3/p4,p5/p6,p5/p7}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-1.75+\drx)*\x},{(6.9+\dry)*\y}) {$\mathbf{xy}$};
		\node[pathnode] at ({(4+\drx)*\x},{(9+\dry)*\y})  {\nuTree{0,1,1,2,1,1,1}{1/0/black,0/1/\blue,1/1/black,2/1/black,3/1/black,4/2/black,0/3/black}{p1/p3,p2/p3,p3/p4,p4/p5,p2/p7}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(4+\drx)*\x},{(8.4+\dry)*\y}) {$\mathbf{xy}$};
		\node[pathnode] at ({(1+\drx)*\x},{(12+\dry)*\y})  {\nuTree{0,1,1,2,1,1,1}{1/0/black,1/1/black,2/1/black,3/1/black,4/2/black,0/3/black}{p1/p2,p2/p3,p3/p4}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(1+\drx)*\x},{(11.4+\dry)*\y}) {$\mathbf{x}$};
		%%
		\draw({(0+\drx)*\x},{(3+\dry)*\y}) node[anchor=west,scale=2]{$\Sigma=x(y{+}1)^{2}$};
		%%
% 		\draw[->](n0310) to[bend right=30] node[midway,below,scale=2] {$\Theta$} (rn0310);
% 		\draw[->](n0310) to[bend left=30] node[midway,below,scale=2] {$\Theta$} (rn0310);
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%%shift facets to the left%%
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%%
		\coordinate (lc1030) at ({(-21-\dlx)*\x},{(0-\dly)*\y});
			\node[fnode,\green] (ln1030) at (lc1030)  {};
		\coordinate (lc1120) at ({(-10.5-\dlx)*\x},{(0-\dly)*\y});
			\node[pathnode,draw,circle] (ln1120) at (lc1120)  {\nuTree{0,1,1,2,1,1,1}{0/0/\blue/,1/0/black/,2/1/black/,3/1/black/,0/2/\blue/,2/2/black/,4/2/black/,0/3/black/}{p1/p2,p1/p5,p3/p4,p3/p6,p5/p6,p5/p8,p6/p7}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-10.5-\dlx)*\x},{(-.6-\dly)*\y}) {$\mathbf{xy^{2}}$};
		\coordinate (lc1021) at ({(-24-\dlx)*\x},{(4-\dly)*\y});
			\node[fnode,\green] (ln1021) at (lc1021)  {};
		\coordinate (lc1111) at ({(-13.5-\dlx)*\x},{(4-\dly)*\y});
			\node[fnode,\green] (ln1111) at (lc1111)  {};
		\coordinate (lc0040) at ({(-21-\dlx)*\x},{(10-\dly)*\y});
			\node[fnode,\green] (ln0040) at (lc0040)  {};
		\coordinate (lc0130) at ({(-10.5-\dlx)*\x},{(10-\dly)*\y});
			\node[fnode,\green] (ln0130) at (lc0130)  {};
		\coordinate (lc0031) at ({(-22.5-\dlx)*\x},{(17-\dly)*\y});
			\node[fnode,\green] (ln0031) at (lc0031)  {};
		\coordinate (lc0121) at ({(-12-\dlx)*\x},{(17-\dly)*\y});
			\node[fnode,\green] (ln0121) at (lc0121)  {};
		\coordinate (lc0022) at ({(-24-\dlx)*\x},{(19-\dly)*\y});
			\node[fnode,\green] (ln0022) at (lc0022)  {};
		\coordinate (lc0112) at ({(-13.5-\dlx)*\x},{(19-\dly)*\y});
			\node[fnode,\green] (ln0112) at (lc0112)  {};
		%%
		\draw[edge,\green,->](ln1030) -- (ln1021);
		\draw[edge,\green,->](ln1030) -- (ln0040);
		\draw[edge,\green,->](ln0121) -- (ln0112);
		\draw[edge,\green,->](ln0112) -- (ln0022);
		\draw[edge,\green,->](ln1021) -- (ln0022);
		\draw[edge,\green,->](ln0130) -- (ln0121);
		\draw[edge,\green,->](ln1120) -- (ln1030);
		\draw[edge,\green,->](ln1120) -- (ln0130);
		\draw[edge,\green,->](ln0040) -- (ln0031);
		\draw[edge,\green,->](ln0031) -- (ln0022);
		\draw[edge,\green,->](ln0121) -- (ln0031);
		\draw[edge,\green,->](ln0130) -- (ln0040);
		\draw[edge,back,\green,->](ln1120) -- (ln1111);
		\draw[edge,back,\green,->](ln1111) -- (ln0112);
		\draw[edge,back,\green,->](ln1111) -- (ln1021);
		%%
		\node[pathnode] at ({(-15.75-\dlx)*\x},{(0-\dly)*\y})  {\nuTree{0,1,1,2,1,1,1}{0/0/\blue/,1/0/black/,3/1/black/,0/2/\blue/,2/2/black/,4/2/black/,0/3/black/}{p1/p2,p1/p4,p4/p5,p4/p7,p5/p6}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-15.75-\dlx)*\x},{(-.6-\dly)*\y}) {$\mathbf{y^{2}}$};
		\node[pathnode] at ({(-10.5-\dlx)*\x},{(5-\dly)*\y})  {\nuTree{0,1,1,2,1,1,1}{1/0/black/,2/1/black/,3/1/black/,0/2/\blue/,2/2/black/,4/2/black/,0/3/black}{p2/p3,p2/p5,p4/p5,p5/p6,p4/p7}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-10.5-\dlx)*\x},{(4.4-\dly)*\y}) {$\mathbf{xy}$};
		\node[pathnode] at ({(-12-\dlx)*\x},{(2-\dly)*\y})  {\nuTree{0,1,1,2,1,1,1}{0/0/\blue/,1/0/black/,2/1/black/,3/1/black/,2/2/black/,4/2/black/,0/3/black/}{p1/p2,p1/p7,p3/p4,p3/p5,p5/p6}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-12-\dlx)*\x},{(1.4-\dly)*\y}) {$\mathbf{xy}$};
		\node[pathnode] at ({(-15.75-\dlx)*\x},{(5-\dly)*\y})  {\nuTree{0,1,1,2,1,1,1}{1/0/black/,3/1/black/,0/2/\blue/,2/2/black/,4/2/black/,0/3/black}{p3/p6,p3/p4,p4/p5}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-15.75-\dlx)*\x},{(4.4-\dly)*\y}) {$\mathbf{y}$};
		\node[pathnode] at ({(-12-\dlx)*\x},{(7-\dly)*\y})  {\nuTree{0,1,1,2,1,1,1}{1/0/black/,2/1/black/,3/1/black/,2/2/black/,4/2/black/,0/3/black}{p2/p3,p2/p4,p4/p5}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-12-\dlx)*\x},{(6.4-\dly)*\y}) {$\mathbf{x}$};
		\node[pathnode] at ({(-17.25-\dlx)*\x},{(2-\dly)*\y})  {\nuTree{0,1,1,2,1,1,1}{0/0/\blue,1/0/black/,3/1/black/,2/2/black/,4/2/black/,0/3/black/}{p1/p2,p1/p6,p4/p5}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-17.25-\dlx)*\x},{(1.4-\dly)*\y}) {$\mathbf{y}$};
		\node[pathnode] at ({(-17.25-\dlx)*\x},{(7-\dly)*\y})  {\nuTree{0,1,1,2,1,1,1}{1/0/black/,3/1/black/,2/2/black/,4/2/black/,0/3/black}{p3/p4}{0/0/\redLabel,0/1/\redLabel,0/2/\redLabel}{.8*\ps}{}{\gray!50!white}};
			\node[anchor=west,scale=\ls] at ({(-17.25-\dlx)*\x},{(6.4-\dly)*\y}) {$\mathbf{1}$};
		%%
		\draw({(-16-\dlx)*\x},{(-2-\dly)*\y}) node[anchor=west,scale=2]{$\Sigma=(x{+}1)(y{+}1)^{2}$};
		%%
% 		\draw[->](n1120) to[bend left=30] node[midway,below,scale=2] {$\Theta$} (ln1120);
	\end{tikzpicture}
\end{page}


%Figure: A $\nu$-associahedron without labels.
\begin{page}%page 11
	\begin{tikzpicture}
		\def\x{1};
		\def\y{1};
		\def\s{.5};
		%%
		\coordinate (c1030) at (-21*\x,0*\y);
			\node[fnode] (n1030) at (c1030)  {};
		\coordinate (c1120) at (-10.5*\x,0*\y);
			\node[fnode,draw,circle] (n1120) at (c1120)  {};
		\coordinate (c1210) at (0*\x,0*\y);
			\node[fnode] (n1210) at (c1210)  {};
		\coordinate (c1021) at (-24*\x,4*\y);
			\node[fnode] (n1021) at (c1021)  {};
		\coordinate (c1111) at (-13.5*\x,4*\y);
			\begin{pgfonlayer}{background}
				\node[fnode] (n1111) at (c1111)  {};
			\end{pgfonlayer}
		\coordinate (c0310) at (0*\x,5*\y);
			\node[fnode,draw,circle] (n0310) at (c0310)  {};
		\coordinate (c1012) at (-25.5*\x,6*\y);
			\node[fnode] (n1012) at (c1012)  {};
		\coordinate (c1003) at (-23.5*\x,8*\y);
			\begin{pgfonlayer}{background}
				\node[fnode] (n1003) at (c1003)  {};
			\end{pgfonlayer}
		\coordinate (c1102) at (-9.5*\x,8*\y);
			\begin{pgfonlayer}{background}
				\node[fnode] (n1102) at (c1102)  {};
			\end{pgfonlayer}
		\coordinate (c1201) at (8*\x,8*\y);
			\node[fnode] (n1201) at (c1201)  {};
		\coordinate (c0040) at (-21*\x,10*\y);
			\node[fnode] (n0040) at (c0040)  {};
		\coordinate (c0130) at (-10.5*\x,10*\y);
			\node[fnode] (n0130) at (c0130)  {};
		\coordinate (c0220) at (-3.5*\x,10*\y);
			\node[fnode] (n0220) at (c0220)  {};
		\coordinate (c0301) at (8*\x,13*\y);
			\node[fnode] (n0301) at (c0301)  {};
		\coordinate (c0031) at (-22.5*\x,17*\y);
			\node[fnode] (n0031) at (c0031)  {};
		\coordinate (c0121) at (-12*\x,17*\y);
			\node[fnode] (n0121) at (c0121)  {};
		\coordinate (c0211) at (-5*\x,17*\y);
			\node[fnode] (n0211) at (c0211)  {};
		\coordinate (c0022) at (-24*\x,19*\y);
			\node[fnode] (n0022) at (c0022)  {};
		\coordinate (c0112) at (-13.5*\x,19*\y);
			\node[fnode] (n0112) at (c0112)  {};
		\coordinate (c0013) at (-25.5*\x,21*\y);
			\node[fnode] (n0013) at (c0013)  {};
		\coordinate (c0004) at (-23.5*\x,23*\y);
			\node[fnode] (n0004) at (c0004)  {};
		\coordinate (c0103) at (-9.5*\x,23*\y);
			\node[fnode] (n0103) at (c0103)  {};
		\coordinate (c0202) at (1*\x,23*\y);
			\node[fnode] (n0202) at (c0202)  {};
		%%
		%% Drawing edges in the back
		%% List of edges in the back
		\def\listEdgesBack{n1003/n0004,n1102/n1003,n1012/n1003,n1201/n1102,n1111/n1102,n1102/n0103,n1120/n1111,n1111/n1021,n1111/n0112};
		\foreach \a/\b in \listEdgesBack{
			\begin{pgfonlayer}{background}
				\draw[edge,back,->] (\a) -- (\b);
			\end{pgfonlayer}
		}
		%% Drawing edges in the front
		%% List of edges in the front
		\def\listEdgesFront{n1210/n1201,n1210/n1120,n1210/n0310,n1201/n0301,n1120/n1030,n1120/n0130,n1030/n1021,n1030/n0040,n1021/n1012,n1021/n0022,n1012/n0013,n0310/n0301,n0310/n0220,n0301/n0202,n0220/n0211,n0220/n0130,n0211/n0202,n0211/n0121,n0202/n0103,n0130/n0121,n0130/n0040,n0121/n0112,n0121/n0031,n0112/n0103,n0112/n0022,n0103/n0004,n0040/n0031,n0031/n0022,n0022/n0013,n0013/n0004};
		%%
		%%
		\foreach \a/\b in \listEdgesFront{
			\draw[edge,->] (\a) -- (\b);
		}
	\end{tikzpicture}
\end{page}

%Figure: A 2-ball
\begin{page}%page 12
	\begin{tikzpicture}\small
		\def\x{2};
		\def\s{.25};
		%%
		\draw(1*\x,1*\x) node[fnode](n1){};
		\draw(2*\x,1*\x) node[fnode](n2){};
		\draw(1.5*\x,1.86*\x) node[fnode](n3){};
		\draw(1.5*\x,1.35*\x) node[fnode](n4){};
		%%
		\draw(n1) -- (n2) -- (n3) -- (n1);
		\draw(n1) -- (n4) -- (n2);
		\draw(n3) -- (n4);
		%%
		\draw(1.5*\x,.9*\x) node[anchor=north,text centered]{$f=(1,4,6,3)$};
		\draw(1.5*\x,.65*\x) node[anchor=north,text centered]{$h=(1,1,1,0)$};
		\draw(1.5*\x,.4*\x) node[anchor=north,text centered]{$\tilde{h}(x)=1+x+x^{2}$};
		\draw(1.5*\x,.15*\x) node[anchor=north,text centered]{$x^{3}\tilde{h}\left(\frac{x+1}{x}\right)=3x^{3}+3x^{2}+x$};
		%%
% 		\begin{pgfonlayer}{background}
% 			\fill[\gray,opacity=.5](1*\x,1*\x) -- (2*\x,1*\x) -- (1.5*\x,1.86*\x) -- (1*\x,1*\x);
% 		\end{pgfonlayer}
% 		%%
		\draw(1.5*\x,2.4*\x) node[anchor=north,text centered]{A subdivided triangle};
		\draw(1.5*\x,2.2*\x) node[anchor=north,text centered]{(a $2$-ball)};
	\end{tikzpicture}
\end{page}

%Figure: A 2-sphere
\begin{page}%page 13
	\begin{tikzpicture}\small
		\def\x{2};
		\def\s{.25};
		%%
		\draw(1*\x,1*\x) node[fnode](n1){};
		\draw(2*\x,1*\x) node[fnode](n2){};
		\draw(1.3*\x,.8*\x) node[fnode](n3){};
		\draw(1.7*\x,1.2*\x) node[fnode](n4){};
		\draw(1.5*\x,.5*\x) node[fnode](n5){};
		\draw(1.5*\x,1.5*\x) node[fnode](n6){};
		%%
		\draw(n1) -- (n3) -- (n2) -- (n6) -- (n1) -- (n5);
		\draw(n6) -- (n3) -- (n5) -- (n2) -- (n4) -- (n6);
		\draw[back](n1) -- (n4) -- (n5);
		%%
		\draw(1.5*\x,.4*\x) node[anchor=north,text centered]{$f=(1,6,12,8)$};
		\draw(1.5*\x,.15*\x) node[anchor=north,text centered]{$h=(1,3,3,1)$};
		\draw(1.5*\x,-.1*\x) node[anchor=north,text centered]{$\tilde{h}(x)=1+3x+3x^{2}+x^{3}$};
		\draw(1.5*\x,-.35*\x) node[anchor=north,text centered]{$x^{3}\tilde{h}\left(\frac{x+1}{x}\right)=8x^{3}+12x^{2}+6x+1$};
		%%
		\draw(1.5*\x,2.25*\x) node[anchor=north,text centered]{The boundary of};
		\draw(1.5*\x,2.05*\x) node[anchor=north,text centered]{an octahedron};
		\draw(1.5*\x,1.85*\x) node[anchor=north,text centered]{(a $2$-sphere)};
	\end{tikzpicture}
\end{page}

%Figure: Three triangles glued along an edge
\begin{page}%page 14
	\begin{tikzpicture}\small
		\def\x{2};
		\def\s{.25};
		%%
		\draw(1.25*\x,1.1*\x) node[fnode](n1){};
		\draw(1.75*\x,1.8*\x) node[fnode](n2){};
		\draw(1.05*\x,1.9*\x) node[fnode](n3){};
		\draw(1.6*\x,.75*\x) node[fnode](n4){};
		\draw(2.15*\x,1.4*\x) node[fnode](n5){};
		\coordinate (isct) at (1.67*\x,1.24*\x);
		%%
		\draw[thick,\blue](n1) -- (n2);
		\draw(n2)-- (n5) -- (n1);
		\draw(n1) -- (n4) -- (isct);
		\draw[back](isct) -- (n2);
		\draw(n1) -- (n3) -- (n2);
		%%
		\draw(1.5*\x,.65*\x) node[anchor=north,text centered]{$f=(1,5,7,3)$};
		\draw(1.5*\x,.4*\x) node[anchor=north,text centered]{$h=(1,2,0,0)$};
		\draw(1.5*\x,.15*\x) node[anchor=north,text centered]{$\tilde{h}(x)=1+2x$};
		\draw(1.5*\x,-.1*\x) node[anchor=north,text centered]{$x^{3}\tilde{h}\left(\frac{x+1}{x}\right)=3x^{3}+2x^{2}$};
		%%
		\draw(1.5*\x,2.35*\x) node[anchor=north,text centered]{Three triangles glued};
		\draw(1.5*\x,2.2*\x) node[anchor=north,text centered]{along an edge};
	\end{tikzpicture}
\end{page}

%Figure: Three tetrahedra glued along an edge
\begin{page}%page 15
	\begin{tikzpicture}\small
		\def\x{2};
		\def\s{.25};
		%%
		\draw(1.2*\x,1.1*\x) node[fnode](n1){};
		\draw(1.6*\x,1.6*\x) node[fnode](n2){};
		\draw(.8*\x,1.4*\x) node[fnode](n3){};
		\draw(1.2*\x,1.9*\x) node[fnode](n4){};
		\draw(2*\x,1.6*\x) node[fnode](n5){};
		\draw(1.9*\x,1*\x) node[fnode](n6){};
		\draw(.95*\x,.6*\x) node[fnode](n7){};
		\draw(1.5*\x,.55*\x) node[fnode](n8){};
		\coordinate (isct) at (1.55*\x,1.06*\x);
		%%
		\draw[thick,\blue](n1) -- (n2);
		\draw(n1) -- (n3) -- (n4) -- (n2);
		\draw(n1) -- (n4);
		\draw[back](n2) -- (n3);
		\draw(n2) -- (n5) -- (n1);
		\draw(n5) -- (n6) -- (n1);
		\draw[back](n2) -- (n6);
		\draw(n1) -- (n7) -- (n8) -- (n1);
		\draw[back](n2) -- (n7);
		\draw(n8) -- (isct);
		\draw[back](isct) -- (n2);
		%%
		\draw(1.5*\x,.45*\x) node[anchor=north,text centered]{$f=(1,8,16,12,3)$};
		\draw(1.5*\x,.2*\x) node[anchor=north,text centered]{$h=(1,4,{\color{red}-2},0,0)$};
		\draw(1.5*\x,-.05*\x) node[anchor=north,text centered]{$\tilde{h}(x)=1+4x-2x^{2}$};
		\draw(1.5*\x,-.3*\x) node[anchor=north,text centered]{$x^{4}\tilde{h}\left(\frac{x+1}{x}\right)=3x^{4}-2x^{2}$};
		%%
		\draw(1.5*\x,2.35*\x) node[anchor=north,text centered]{Three tetrahedra glued};
		\draw(1.5*\x,2.2*\x) node[anchor=north,text centered]{along an edge};
	\end{tikzpicture}
\end{page}

%Figure: A triangulated cylinder
\begin{page}%page 16
	\begin{tikzpicture}\small
		\def\x{.5};
		\def\s{.25};
		%%
		\draw(.5*\x,.5*\x) node{};
		\draw(6.5*\x,4.5*\x) node{};
		%%
		\draw(1*\x,1*\x) node[fnode](n1){};
		\draw(6*\x,1*\x) node[fnode](n2){};
		\draw(1*\x,4*\x) node[fnode](n3){};
		\draw(6*\x,4*\x) node[fnode](n4){};
		%%
		\draw(n1) -- (n2);
		\draw(n2) -- (n4);
		\draw(n2) -- (n3);
		\draw(n4) -- (n3);
		\draw(n1) -- (n3);
		%%
		\draw[<->](6.5*\x,1.25*\x) to[bend right=65] (6.5*\x,3.75*\x);
		\draw(3.25*\x,1.15*\x) -- (3.55*\x,.85*\x);
		\draw(3.45*\x,1.15*\x) -- (3.75*\x,.85*\x);
		\draw(3.25*\x,4.15*\x) -- (3.55*\x,3.85*\x);
		\draw(3.45*\x,4.15*\x) -- (3.75*\x,3.85*\x);
		%%
		\draw(10*\x,4*\x) node[fnode]{};
		\draw(15*\x,4*\x) node[fnode]{};
		%%
		\draw[rounded corners](10*\x,1*\x) -- (15*\x,1*\x);
		\draw[rounded corners](10*\x,4*\x) -- (15*\x,4*\x);
		\draw(15.5*\x,2.5*\x) arc (0:360:0.5*\x cm and 1.5*\x cm);
		\draw[back](10*\x,1*\x) arc (-90:90:0.5*\x cm and 1.5*\x cm);
		\draw(10*\x,4*\x) arc (90:270:0.5*\x cm and 1.5*\x cm);
		%%
		\draw(10*\x,4*\x) arc(90:65:0.5*\x cm and 1.5*\x cm) -- (12.25*\x,1.2*\x);
		\draw(12.25*\x,1.2*\x) arc(240:270:.5*\x cm and 1.5*\x cm);
		\draw[back](12.5*\x,1*\x) arc(270:300:.5*\x cm and 1.5*\x cm) -- (14.75*\x,3.8*\x);
		\draw[back](15*\x,4*\x) arc(90:115:0.5*\x cm and 1.5*\x cm);
	\end{tikzpicture}
\end{page}

%Figure: Another triangulated cylinder
\begin{page}%page 17
	\begin{tikzpicture}\small
		\def\x{.5};
		\def\s{.25};
		%%
		\draw(.5*\x,.5*\x) node{};
		\draw(6.5*\x,5*\x) node{};
		%%
		\draw(1*\x,.5*\x) node[fnode](n1){};
			\draw(.8*\x,.5*\x) node[anchor=east,scale=.8]{$a$};
		\draw(6*\x,.5*\x) node[fnode](n2){};
			\draw(6.2*\x,.5*\x) node[anchor=west,scale=.8]{$b$};
		\draw(1*\x,2.5*\x) node[fnode](n3){};
			\draw(.8*\x,2.5*\x) node[anchor=east,scale=.8]{$c$};
		\draw(6*\x,2.5*\x) node[fnode](n4){};
			\draw(6.2*\x,2.5*\x) node[anchor=west,scale=.8]{$d$};
		\draw(1*\x,4.5*\x) node[fnode](n5){};
			\draw(.8*\x,4.5*\x) node[anchor=east,scale=.8]{$a$};
		\draw(6*\x,4.5*\x) node[fnode](n6){};
			\draw(6.2*\x,4.5*\x) node[anchor=west,scale=.8]{$b$};
		%%
		\draw(n1) -- (n2);
		\draw(n1) -- (n3);
		\draw(n2) -- (n3);
		\draw(n2) -- (n4);
		\draw[thick,\blue](n3) -- (n4);
		\draw(n3) -- (n5);
		\draw(n4) -- (n5);
		\draw(n4) -- (n6);
		\draw(n5) -- (n6);
		%%
% 		\draw[<->](6.5*\x,1.25*\x) to[bend right=65] (6.5*\x,4.75*\x);
		\draw(3.25*\x,.65*\x) -- (3.55*\x,.35*\x);
		\draw(3.45*\x,.65*\x) -- (3.75*\x,.35*\x);
		\draw(3.25*\x,4.65*\x) -- (3.55*\x,4.35*\x);
		\draw(3.45*\x,4.65*\x) -- (3.75*\x,4.35*\x);
		%%
		\draw(10*\x,1*\x) node(p1)[fnode]{};
			\draw(9.8*\x,1*\x) node[anchor=east,scale=.8]{$c$};
		\draw(15*\x,1*\x) node(p2)[fnode]{};
			\draw(15.2*\x,1*\x) node[anchor=west,scale=.8]{$d$};
		\draw(10*\x,4*\x) node(p3)[fnode]{};
			\draw(9.8*\x,4*\x) node[anchor=east,scale=.8]{$a$};
		\draw(15*\x,4*\x) node(p4)[fnode]{};
			\draw(15.2*\x,4*\x) node[anchor=west,scale=.8]{$b$};
		%%
		\draw[rounded corners,thick,\blue](p1) -- (p2);
		\draw[rounded corners](p3) -- (p4);
		\draw(12.25*\x,4.15*\x) -- (12.55*\x,3.85*\x);
		\draw(12.45*\x,4.15*\x) -- (12.75*\x,3.85*\x);
		%%
		\draw(10.5*\x,2.5*\x) arc (0:360:0.5*\x cm and 1.5*\x cm);
		\draw(15*\x,1*\x) arc (-90:90:0.5*\x cm and 1.5*\x cm);
		\draw[back](15*\x,4*\x) arc (90:270:0.5*\x cm and 1.5*\x cm);
		%%
		\draw(p3) to[bend left=15] (p2);
		\draw[back](p1) to[bend right=15] (p4);
	\end{tikzpicture}
\end{page}

%Figure: A labeled poset.
\begin{page}%page 18
		\begin{tikzpicture}\small
		\def\x{1};
		\def\y{1};
		\def\s{.6};
		%%
		\draw(2*\x,1*\y) node[fnode](n1){};
		\draw(4.5*\x,1*\y) node[fnode](n2){};
		\draw(1*\x,2*\y) node[fnode](n3){};
		\draw(2*\x,2*\y) node[fnode,draw=black,fill=\red](n4){};
		\draw(3*\x,2*\y) node[fnode](n5){};
		\draw(4*\x,2*\y) node[fnode](n6){};
		\draw(5*\x,2*\y) node[fnode](n7){};
		\draw(1*\x,3*\y) node[fnode,draw=black,fill=\blue!50!white](n8){};
		\draw(2*\x,3*\y) node[fnode,draw=black,fill=\blueLabel](n9){};
		\draw(5*\x,3*\y) node[fnode](n10){};
		\draw(1*\x,4*\y) node[fnode](n11){};
		\draw(4*\x,4*\y) node[fnode](n12){};
		\draw(2*\x,5*\y) node[fnode](n13){};
		%%
		\draw[edge](n1) -- (n3) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{0}$};
		\draw[edge](n1) -- (n4) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{1}$};
		\draw[edge](n1) -- (n5) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{0}$};
		\draw[edge](n2) -- (n4) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{2}$};
		\draw[edge](n2) -- (n5) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{1}$};
		\draw[edge](n2) -- (n6) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{0}$};
		\draw[edge](n2) -- (n7) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{0}$};
		\draw[edge](n4) -- (n8) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{2}$};
		\draw[edge](n4) -- (n9) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{1}$};
		\draw[edge](n5) -- (n12) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{0}$};
		\draw[edge](n7) -- (n10) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{0}$};
		\draw[edge](n7) -- (n13) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{0}$};
		\draw[edge](n8) -- (n11) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{1}$};
		\draw[edge](n9) -- (n11) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{1}$};
		\draw[edge](n10) -- (n12) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{2}$};
		\draw[edge](n11) -- (n13) node[midway,fill=white,inner sep=1pt,scale=.75]{$\mathbf{2}$};
	\end{tikzpicture}
\end{page}


\end{document}
