% % THIS DOCUMENT IS WRITTEN IN LATEX % % TO FIND THE TITLE: search for the command \title using your wordprocessor % \documentclass[titlepage,12pt]{article} %[12pt]{amsart} \usepackage{amssymb} \usepackage{latexsym} %\usepackage{amstex} %\renewcommand{\baselinestretch}{2} \newcommand{\ben}{\begin{enumerate}} \newcommand{\een}{\end{enumerate}} \newcommand{\ble}{\begin{lem}} \newcommand{\ele}{\end{lem}} \newcommand{\bth}{\begin{thm}} \renewcommand{\eth}{\end{thm}} \newcommand{\bpr}{\begin{prop}} \newcommand{\epr}{\end{prop}} \newcommand{\bco}{\begin{cor}} \newcommand{\eco}{\end{cor}} \newcommand{\bcon}{\begin{conj}} \newcommand{\econ}{\end{conj}} \newcommand{\bde}{\begin{defn}} \newcommand{\ede}{\end{defn}} \newcommand{\bex}{\begin{exa}} \newcommand{\eex}{\end{exa}} \newcommand{\barr}{\begin{array}} \newcommand{\earr}{\end{array}} \newcommand{\btab}{\begin{tabular}} \newcommand{\etab}{\end{tabular}} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bea}{\begin{eqnarray*}} \newcommand{\eea}{\end{eqnarray*}} \newcommand{\bce}{\begin{center}} \newcommand{\ece}{\end{center}} \newcommand{\bpi}{\begin{picture}} \newcommand{\epi}{\end{picture}} \newcommand{\bfi}{\begin{figure} \begin{center}} \newcommand{\efi}{\end{center} \end{figure}} \newcommand{\capt}{\caption} \newcommand{\bsl}{\begin{slide}{}} \newcommand{\esl}{\end{slide}} \newcommand{\bib}{thebibliography} \newcommand{\pf}{{\bf Proof.}} \newcommand{\qed}{\rule{1ex}{1ex}} \newcommand{\Qed}{\rule{1ex}{1ex} \medskip} \newcommand{\mc}[2]{\multicolumn{#1}{#2}} \newcommand{\ul}{\underline} \newcommand{\ol}{\overline} \newcommand{\hs}[1]{\hspace{#1}} \newcommand{\vs}[1]{\vspace{#1}} \newcommand{\ssk}{\smallskip} \newcommand{\msk}{\mediumskip} \newcommand{\bsk}{\bigskip} \newcommand{\rp}[2]{\rule{#1pt}{#2pt}} \newcommand{\st}[1]{\rule{#1}{0pt}} \newcommand{\mbl}[1]{\makebox(0,0)[l]{#1}} \newcommand{\mbr}[1]{\makebox(0,0)[r]{#1}} \newcommand{\mbt}[1]{\makebox(0,0)[t]{#1}} \newcommand{\mbb}[1]{\makebox(0,0)[b]{#1}} \newcommand{\mbc}[1]{\makebox(0,0){#1}} \newcommand{\emp}{\emptyset} \newcommand{\sbs}{\subset} \newcommand{\sbe}{\subseteq} \newcommand{\sps}{\supset} \newcommand{\spe}{\supseteq} \newcommand{\setm}{\setminus} \newcommand{\asy}{\thicksim} \newcommand{\dle}{<\hspace{-6pt}\cdot} \newcommand{\dge}{\cdot\hspace{-6pt}>} \newcommand{\iso}{\cong} \newcommand{\mh}{\hat{0}} \newcommand{\Mh}{\hat{1}} \newcommand{\Dh}{\hat{D}} \newcommand{\zh}{\hat{0}} \newcommand{\oh}{\hat{1}} \newcommand{\Ph}{\hat{P}} \newcommand{\Pih}{\hat{\Pi}} \newcommand{\Sh}{\hat{S}} \newcommand{\lh}{\hat{l}} \newcommand{\xh}{\hat{x}} \newcommand{\lt}{\lhd} \newcommand{\gt}{\rhd} \newcommand{\lte}{\unlhd} \newcommand{\gte}{\unrhd} \newcommand{\jn}{\vee} \newcommand{\Jn}{\bigvee} \newcommand{\mt}{\wedge} \newcommand{\Mt}{\bigwedge} \newcommand{\bdy}{\partial} \newcommand{\sd}{\bigtriangleup} \newcommand{\case}[4]{\left\{\barr{ll}#1&\mbox{#2}\\#3&\mbox{#4}\earr\right.} \newcommand{\fl}[1]{\lfloor #1 \rfloor} \newcommand{\ce}[1]{\lceil #1 \rceil} \newcommand{\flf}[2]{\left\lfloor\frac{#1}{#2}\right\rfloor} \newcommand{\cef}[2]{\left\lceil\frac{#1}{#2}\right\rceil} \newcommand{\gau}[2]{\left[ #1 \atop #2 \right]} \newcommand{\setc}[2]{\{left{#1}\ :\ #2\}} \newcommand{\setl}[2]{\{left{#1}\ |\ #2\}} \newcommand{\qst}{$q$-Stirling numbers} \newcommand{\qbi}{$q$-binomial coefficients} \newcommand{\del}{\nabla} \newcommand{\pde}[2]{\frac{\partial#1}{\partial#2}} \newcommand{\spd}[2]{\frac{\partial^2\hspace{-2pt}#1}{\partial#2^2}} \newcommand{\ode}[2]{\ds\frac{d#1}{d#2}} \newcommand{\fall}[2]{\langle{#1}\rangle_{#2}} \newcommand{\nor}[1]{\parallel{#1}\parallel} \newcommand{\ipr}[1]{\langle{#1}\rangle} \newcommand{\lb}{\left\{} \newcommand{\rb}{\right\}} \newcommand{\ree}[1]{(\ref{#1})} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\rla}{\leftrightarrow} \newcommand{\lra}{\longrightarrow} \newcommand{\lla}{\longleftarrow} \newcommand{\Lra}{\Longrightarrow} \newcommand{\llra}{\longleftrightarrow} \newcommand{\Llra}{\Longleftrightarrow} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\de}{\delta} \newcommand{\ep}{\epsilon} \newcommand{\io}{\iota} \newcommand{\la}{\lambda} \newcommand{\mut}{\tilde{\mu}} \newcommand{\om}{\omega} \newcommand{\si}{\sigma} \renewcommand{\th}{\theta} % NOTE THAT \th HAS BEEN RENEWCOMMANDed \newcommand{\ze}{\zeta} \newcommand{\Al}{\Alpha} \newcommand{\Ga}{\Gamma} \newcommand{\De}{\Delta} \newcommand{\Ep}{\epsilon} \newcommand{\La}{\Lambda} \newcommand{\Om}{\Omega} \newcommand{\Si}{\Sigma} \newcommand{\Th}{\Theta} \newcommand{\Ze}{\Zeta} \newcommand{\ba}{{\bf a}} \newcommand{\bb}{{\bf b}} \newcommand{\bc}{{\bf c}} \newcommand{\bd}{{\bf d}} \newcommand{\bi}{{\bf i}} \newcommand{\bj}{{\bf j}} \newcommand{\bk}{{\bf k}} \newcommand{\bm}{{\bf m}} \newcommand{\bn}{{\bf n}} \newcommand{\bp}{{\bf p}} \newcommand{\bq}{{\bf q}} \newcommand{\br}{{\bf r}} \newcommand{\bs}{{\bf s}} \newcommand{\bt}{{\bf t}} \newcommand{\bu}{{\bf u}} \newcommand{\bv}{{\bf v}} \newcommand{\bw}{{\bf w}} \newcommand{\bx}{{\bf x}} \newcommand{\by}{{\bf y}} \newcommand{\bA}{{\bf A}} \newcommand{\bB}{{\bf B}} \newcommand{\bC}{{\bf C}} \newcommand{\bE}{{\bf E}} \newcommand{\bF}{{\bf F}} \newcommand{\bN}{{\bf N}} \newcommand{\bP}{{\bf P}} \newcommand{\bQ}{{\bf Q}} \newcommand{\bR}{{\bf R}} \newcommand{\bT}{{\bf T}} \newcommand{\bX}{{\bf X}} \newcommand{\bXh}{\hat{{\bf X}}} \newcommand{\bZ}{{\bf Z}} \newcommand{\bbC}{{\mathbb C}} \newcommand{\bbF}{{\mathbb F}} \newcommand{\bbN}{{\mathbb N}} \newcommand{\bbR}{{\mathbb R}} \newcommand{\bbZ}{{\mathbb Z}} \newcommand{\cA}{{\cal A}} \newcommand{\cB}{{\cal B}} \newcommand{\cC}{{\cal C}} \newcommand{\cD}{{\cal D}} \newcommand{\cF}{{\cal F}} \newcommand{\cG}{{\cal G}} \newcommand{\cI}{{\cal I}} \newcommand{\cJ}{{\cal J}} \newcommand{\cL}{{\cal L}} \newcommand{\cO}{{\cal O}} \newcommand{\cP}{{\cal P}} \newcommand{\cR}{{\cal R}} \newcommand{\cS}{{\cal S}} \newcommand{\cT}{{\cal T}} \newcommand{\cW}{{\cal W}} \newcommand{\cAB}{{\cal AB}} \newcommand{\cBC}{{\cal BC}} \newcommand{\cDB}{{\cal DB}} \newcommand{\cAD}{{\cal AD}} \newcommand{\cIF}{{\cal IF}} \newcommand{\cNBC}{{\cal NBC}} \newcommand{\cRF}{{\cal RF}} \newcommand{\df}{\dot{f}} \newcommand{\dF}{\dot{F}} \newcommand{\fS}{{\frak S}} \newcommand{\at}{\tilde{a}} \newcommand{\ct}{\tilde{c}} \newcommand{\dt}{\tilde{d}} \newcommand{\Bt}{\tilde{B}} \newcommand{\Gt}{\tilde{G}} \newcommand{\Ht}{\tilde{H}} \newcommand{\Kt}{\tilde{K}} \newcommand{\Nt}{\tilde{N}} \newcommand{\St}{\tilde{S}} \newcommand{\alt}{\tilde{\alpha}} \newcommand{\bet}{\tilde{\beta}} \newcommand{\rht}{\tilde{\rho}} \newcommand{\Pit}{\tilde{\Pi}} \newcommand{\tal}{\tilde{\alpha}} \newcommand{\tbe}{\tilde{\beta}} \newcommand{\tPi}{\tilde{\Pi}} \newcommand{\ab}{\ol{a}} \newcommand{\Bb}{\ol{B}} \newcommand{\cb}{\ol{c}} \newcommand{\Cb}{\ol{C}} \newcommand{\db}{\ol{d}} \newcommand{\Db}{\ol{D}} \newcommand{\eb}{\ol{e}} \newcommand{\fb}{\ol{f}} \newcommand{\Gb}{\ol{G}} \newcommand{\Hb}{\ol{H}} \newcommand{\kb}{\ol{k}} \newcommand{\Kb}{\ol{K}} \newcommand{\nb}{\ol{n}} \newcommand{\Phib}{\ol{\Phi}} %\newcommand{\Sb}{\ol{S}} \newcommand{\Wb}{\ol{W}} \newcommand{\yb}{\ol{y}} \newcommand{\Yb}{\ol{Y}} \newcommand{\pib}{\ol{\pi}} \newcommand{\sib}{\ol{\si}} \newcommand{\degb}{\ol{\deg}} \newcommand{\vj}{\vec{\jmath}} \newcommand{\vv}{\vec{v}} \newcommand{\Aut}{\mathop{\rm Aut}} \newcommand{\abl}{\mathop {\rm al}} \newcommand{\codim}{\mathop{\rm codim}} \newcommand{\col}{\mathop{\rm col}} \newcommand{\Der}{\mathop{\rm Der}} \newcommand{\des}{\mathop{\rm des}} \newcommand{\Des}{\mathop{\rm Des}} \newcommand{\diag}{\mathop{\rm diag}} \newcommand{\diam}{\mathop{\rm diam}} \newcommand{\ev}{\mathop{\rm ev}} \newcommand{\eval}{\mathop{\rm eval}} \newcommand{\Eval}{\mathop{\rm Eval}} \newcommand{\FPF}{\mathop{\rm FPF}} \newcommand{\Harm}{{\rm Harm}} \newcommand{\id}{\mathop {\rm id}} \newcommand{\Ind}{\mathop {\rm Ind}} \newcommand{\inter}{\mathop {\rm int}} \newcommand{\Inter}{\mathop {\rm Int}} \newcommand{\inv}{\mathop {\rm inv}} \newcommand{\Inv}{\mathop {\rm Inv}} \newcommand{\lcm}{\mathop {\rm lcm}} \newcommand{\LHS}{\mathop {\rm LHS}} \newcommand{\Mat}{\mathop {\rm Mat}} \newcommand{\maj}{\mathop {\rm maj}} %\newcommand{\mod}{\mathop{\rm mod}} \newcommand{\nin}{\mathop {\rm nin}} \newcommand{\Nin}{\mathop {\rm Nin}} \newcommand{\nul}{\mathop {\rm nul}} \newcommand{\od}{\mathop {\rm od}} \newcommand{\pfa}{\mathop {\rm pf}} \newcommand{\Pm}{\mathop {\rm pm}} \newcommand{\PM}{\mathop {\rm PM}} \newcommand{\Pol}{{\rm Pol}} \newcommand{\RHS}{\mathop {\rm RHS}} \newcommand{\rk}{\mathop{\rm rk}} \newcommand{\rank}{\mathop{\rm rank}} \newcommand{\row}{\mathop{\rm row}} \newcommand{\sdiam}{\mathop{\rm sdiam}} \newcommand{\sgn}{\mathop{\rm sgn}} \newcommand{\slack}{{\rm slack}} \newcommand{\summ}{\mathop{\rm sum}} \newcommand{\tr}{\mathop{\rm tr}} \newcommand{\wt}{\mathop {\rm wt}} \newcommand{\wed}{\mathop {\rm wedge}} \newcommand{\sus}{\mathop {\rm susp}} \newcommand{\Sym}{\mathop {\rm Sym}} \newcommand{\dil}{\displaystyle} \newcommand{\scl}{\scriptstyle} \newcommand{\ssl}{\scriptscriptstyle} \newcommand{\foz}{\footnotesize} \newcommand{\scz}{\scriptsize} \newcommand{\cho}{\choose} \newcommand{\Schu}{Sch\"utzenberger} \newcommand{\aim}{Adv. in Math.} \newcommand{\bams}{Bull. Amer. Math. Soc.} \newcommand{\cjm}{Canad. J. Math.} \newcommand{\dm}{Discrete Math.} \newcommand{\dmj}{Duke Math. J.} \newcommand{\ejc}{European J. Combin.} \newcommand{\im}{Invent. Math.} \newcommand{\jaa}{J. Algebra} \newcommand{\jac}{J. Algebraic Combin.} \newcommand{\jas}{J. Algorithms} \newcommand{\jams}{J. Amer. Math. Soc.} \newcommand{\jct}{J. Combin. Theory} \newcommand{\jcta}{J. Combin. Theory Ser. A} \newcommand{\jctb}{J. Combin. Theory Ser. B} \newcommand{\jgt}{J. Graph Theory} \newcommand{\jram}{J. Reine Angew. Math.} \newcommand{\pjm}{Pacific J. Math.} \newcommand{\pams}{Proc. Amer. Math. Soc.} \newcommand{\plms}{Proc. London Math. Soc.} \newcommand{\tams}{Trans. Amer. Math. Soc.} \newcommand{\pja}{Proc. Japan Acad. Ser. A Math} \newcommand{\sv}{Springer-Verlag Lecture Notes in Math.} \newcommand{\crgs}{Combinatoire et Repr\'{e}sentation du Groupe Sym\'{e}trique, Strasbourg 1976, D. Foata ed.} \newcommand{\caup}{Cambridge University Press} \newcommand{\oup}{Oxford University Press} \newcommand{\pr}{preprint} \newcommand{\ip}{in preparation} \newcommand{\ds}{\displaystyle} \setlength{\topmargin}{.1in} \setlength{\textheight}{8in} \setlength{\textwidth}{5.8in} \setlength{\evensidemargin}{.4in} \setlength{\oddsidemargin}{.4in} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{conj}[thm]{Conjecture} \newtheorem{exa}[thm]{Example} \newtheorem{defn}[thm]{Definition} % % THE GRID - UNLABELED VERTICES % \newcommand{\Gaa}{\put(0,0){\circle*{3}}} \newcommand{\Gba}{\put(10,0){\circle*{3}}} \newcommand{\Gca}{\put(20,0){\circle*{3}}} \newcommand{\Gda}{\put(30,0){\circle*{3}}} \newcommand{\Gea}{\put(40,0){\circle*{3}}} \newcommand{\Gfa}{\put(50,0){\circle*{3}}} \newcommand{\Gga}{\put(60,0){\circle*{3}}} \newcommand{\Gab}{\put(0,10){\circle*{3}}} \newcommand{\Gbb}{\put(10,10){\circle*{3}}} \newcommand{\Gcb}{\put(20,10){\circle*{3}}} \newcommand{\Gdb}{\put(30,10){\circle*{3}}} \newcommand{\Geb}{\put(40,10){\circle*{3}}} \newcommand{\Gfb}{\put(50,10){\circle*{3}}} \newcommand{\Ggb}{\put(60,10){\circle*{3}}} \newcommand{\Gac}{\put(0,20){\circle*{3}}} \newcommand{\Gbc}{\put(10,20){\circle*{3}}} \newcommand{\Gcc}{\put(20,20){\circle*{3}}} \newcommand{\Gdc}{\put(30,20){\circle*{3}}} \newcommand{\Gec}{\put(40,20){\circle*{3}}} \newcommand{\Gfc}{\put(50,20){\circle*{3}}} \newcommand{\Ggc}{\put(60,20){\circle*{3}}} \newcommand{\Gad}{\put(0,30){\circle*{3}}} \newcommand{\Gbd}{\put(10,30){\circle*{3}}} \newcommand{\Gcd}{\put(20,30){\circle*{3}}} \newcommand{\Gdd}{\put(30,30){\circle*{3}}} \newcommand{\Ged}{\put(40,30){\circle*{3}}} \newcommand{\Gfd}{\put(50,30){\circle*{3}}} \newcommand{\Ggd}{\put(60,30){\circle*{3}}} \newcommand{\Gae}{\put(0,40){\circle*{3}}} \newcommand{\Gbe}{\put(10,40){\circle*{3}}} \newcommand{\Gce}{\put(20,40){\circle*{3}}} \newcommand{\Gde}{\put(30,40){\circle*{3}}} \newcommand{\Gee}{\put(40,40){\circle*{3}}} \newcommand{\Gfe}{\put(50,40){\circle*{3}}} \newcommand{\Gge}{\put(60,40){\circle*{3}}} \newcommand{\Gaf}{\put(0,50){\circle*{3}}} \newcommand{\Gbf}{\put(10,50){\circle*{3}}} \newcommand{\Gcf}{\put(20,50){\circle*{3}}} \newcommand{\Gdf}{\put(30,50){\circle*{3}}} \newcommand{\Gef}{\put(40,50){\circle*{3}}} \newcommand{\Gff}{\put(50,50){\circle*{3}}} \newcommand{\Ggf}{\put(60,50){\circle*{3}}} \newcommand{\Gag}{\put(0,60){\circle*{3}}} \newcommand{\Gbg}{\put(10,60){\circle*{3}}} \newcommand{\Gcg}{\put(20,60){\circle*{3}}} \newcommand{\Gdg}{\put(30,60){\circle*{3}}} \newcommand{\Geg}{\put(40,60){\circle*{3}}} \newcommand{\Gfg}{\put(50,60){\circle*{3}}} \newcommand{\Ggg}{\put(60,60){\circle*{3}}} % % THE GRID - UNLABELED VERTICES, VARIABLE SIZE % \newcommand{\GaaV}[1]{\put(0,0){\circle*{#1}}} \newcommand{\GbaV}[1]{\put(10,0){\circle*{#1}}} \newcommand{\GcaV}[1]{\put(20,0){\circle*{#1}}} \newcommand{\GdaV}[1]{\put(30,0){\circle*{#1}}} \newcommand{\GeaV}[1]{\put(40,0){\circle*{#1}}} \newcommand{\GfaV}[1]{\put(50,0){\circle*{#1}}} \newcommand{\GgaV}[1]{\put(60,0){\circle*{#1}}} \newcommand{\GabV}[1]{\put(0,10){\circle*{#1}}} \newcommand{\GbbV}[1]{\put(10,10){\circle*{#1}}} \newcommand{\GcbV}[1]{\put(20,10){\circle*{#1}}} \newcommand{\GdbV}[1]{\put(30,10){\circle*{#1}}} \newcommand{\GebV}[1]{\put(40,10){\circle*{#1}}} \newcommand{\GfbV}[1]{\put(50,10){\circle*{#1}}} \newcommand{\GgbV}[1]{\put(60,10){\circle*{#1}}} \newcommand{\GacV}[1]{\put(0,20){\circle*{#1}}} \newcommand{\GbcV}[1]{\put(10,20){\circle*{#1}}} \newcommand{\GccV}[1]{\put(20,20){\circle*{#1}}} \newcommand{\GdcV}[1]{\put(30,20){\circle*{#1}}} \newcommand{\GecV}[1]{\put(40,20){\circle*{#1}}} \newcommand{\GfcV}[1]{\put(50,20){\circle*{#1}}} \newcommand{\GgcV}[1]{\put(60,20){\circle*{#1}}} \newcommand{\GadV}[1]{\put(0,30){\circle*{#1}}} \newcommand{\GbdV}[1]{\put(10,30){\circle*{#1}}} \newcommand{\GcdV}[1]{\put(20,30){\circle*{#1}}} \newcommand{\GddV}[1]{\put(30,30){\circle*{#1}}} \newcommand{\GedV}[1]{\put(40,30){\circle*{#1}}} \newcommand{\GfdV}[1]{\put(50,30){\circle*{#1}}} \newcommand{\GgdV}[1]{\put(60,30){\circle*{#1}}} \newcommand{\GaeV}[1]{\put(0,40){\circle*{#1}}} \newcommand{\GbeV}[1]{\put(10,40){\circle*{#1}}} \newcommand{\GceV}[1]{\put(20,40){\circle*{#1}}} \newcommand{\GdeV}[1]{\put(30,40){\circle*{#1}}} \newcommand{\GeeV}[1]{\put(40,40){\circle*{#1}}} \newcommand{\GfeV}[1]{\put(50,40){\circle*{#1}}} \newcommand{\GgeV}[1]{\put(60,40){\circle*{#1}}} \newcommand{\GafV}[1]{\put(0,50){\circle*{#1}}} \newcommand{\GbfV}[1]{\put(10,50){\circle*{#1}}} \newcommand{\GcfV}[1]{\put(20,50){\circle*{#1}}} \newcommand{\GdfV}[1]{\put(30,50){\circle*{#1}}} \newcommand{\GefV}[1]{\put(40,50){\circle*{#1}}} \newcommand{\GffV}[1]{\put(50,50){\circle*{#1}}} \newcommand{\GgfV}[1]{\put(60,50){\circle*{#1}}} \newcommand{\GagV}[1]{\put(0,60){\circle*{#1}}} \newcommand{\GbgV}[1]{\put(10,60){\circle*{#1}}} \newcommand{\GcgV}[1]{\put(20,60){\circle*{#1}}} \newcommand{\GdgV}[1]{\put(30,60){\circle*{#1}}} \newcommand{\GegV}[1]{\put(40,60){\circle*{#1}}} \newcommand{\GfgV}[1]{\put(50,60){\circle*{#1}}} \newcommand{\GggV}[1]{\put(60,60){\circle*{#1}}} % % THE GRID - LABLED VERTICES % \newcommand{\GaaL}[2]{\Gaa \put(0,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbaL}[2]{\Gba \put(10,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcaL}[2]{\Gca \put(20,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdaL}[2]{\Gda \put(30,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GeaL}[2]{\Gea \put(40,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfaL}[2]{\Gfa \put(50,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgaL}[2]{\Gga \put(60,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GabL}[2]{\Gab \put(0,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbbL}[2]{\Gbb \put(10,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcbL}[2]{\Gcb \put(20,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdbL}[2]{\Gdb \put(30,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GebL}[2]{\Geb \put(40,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfbL}[2]{\Gfb \put(50,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgbL}[2]{\Ggb \put(60,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GacL}[2]{\Gac \put(0,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbcL}[2]{\Gbc \put(10,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GccL}[2]{\Gcc \put(20,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdcL}[2]{\Gdc \put(30,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GecL}[2]{\Gec \put(40,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfcL}[2]{\Gfc \put(50,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgcL}[2]{\Ggc \put(60,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GadL}[2]{\Gad \put(0,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbdL}[2]{\Gbd \put(10,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcdL}[2]{\Gcd \put(20,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GddL}[2]{\Gdd \put(30,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GedL}[2]{\Ged \put(40,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfdL}[2]{\Gfd \put(50,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgdL}[2]{\Ggd \put(60,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GaeL}[2]{\Gae \put(0,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbeL}[2]{\Gbe \put(10,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GceL}[2]{\Gce \put(20,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdeL}[2]{\Gde \put(30,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GeeL}[2]{\Gee \put(40,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfeL}[2]{\Gfe \put(50,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgeL}[2]{\Gge \put(60,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GafL}[2]{\Gaf \put(0,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbfL}[2]{\Gbf \put(10,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcfL}[2]{\Gcf \put(20,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdfL}[2]{\Gdf \put(30,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GefL}[2]{\Gef \put(40,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GffL}[2]{\Gff \put(50,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgfL}[2]{\Ggf \put(60,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GagL}[2]{\Gag \put(0,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbgL}[2]{\Gbg \put(10,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcgL}[2]{\Gcg \put(20,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdgL}[2]{\Gdg \put(30,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GegL}[2]{\Geg \put(40,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfgL}[2]{\Gfg \put(50,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GggL}[2]{\Ggg \put(60,60){\makebox(0,0){\hspace{#1}#2}}} % % THE GRID - LABLED VERTICES, VARIABLE SIZE % \newcommand{\GaaLV}[3]{\GaaV{#3} \put(0,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbaLV}[3]{\GbaV{#3} \put(10,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcaLV}[3]{\GcaV{#3} \put(20,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdaLV}[3]{\GdaV{#3} \put(30,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GeaLV}[3]{\GeaV{#3} \put(40,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfaLV}[3]{\GfaV{#3} \put(50,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgaLV}[3]{\GgaV{#3} \put(60,0){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GabLV}[3]{\GabV{#3} \put(0,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbbLV}[3]{\GbbV{#3} \put(10,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcbLV}[3]{\GcbV{#3} \put(20,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdbLV}[3]{\GdbV{#3} \put(30,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GebLV}[3]{\GebV{#3} \put(40,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfbLV}[3]{\GfbV{#3} \put(50,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgbLV}[3]{\GgbV{#3} \put(60,10){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GacLV}[3]{\GacV{#3} \put(0,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbcLV}[3]{\GbcV{#3} \put(10,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GccLV}[3]{\GccV{#3} \put(20,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdcLV}[3]{\GdcV{#3} \put(30,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GecLV}[3]{\GecV{#3} \put(40,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfcLV}[3]{\GfcV{#3} \put(50,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgcLV}[3]{\GgcV{#3} \put(60,20){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GadLV}[3]{\GadV{#3} \put(0,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbdLV}[3]{\GbdV{#3} \put(10,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcdLV}[3]{\GcdV{#3} \put(20,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GddLV}[3]{\GddV{#3} \put(30,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GedLV}[3]{\GedV{#3} \put(40,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfdLV}[3]{\GfdV{#3} \put(50,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgdLV}[3]{\GgdV{#3} \put(60,30){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GaeLV}[3]{\GaeV{#3} \put(0,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbeLV}[3]{\GbeV{#3} \put(10,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GceLV}[3]{\GceV{#3} \put(20,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdeLV}[3]{\GdeV{#3} \put(30,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GeeLV}[3]{\GeeV{#3} \put(40,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfeLV}[3]{\GfeV{#3} \put(50,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgeLV}[3]{\GgeV{#3} \put(60,40){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GafLV}[3]{\GafV{#3} \put(0,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbfLV}[3]{\GbfV{#3} \put(10,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcfLV}[3]{\GcfV{#3} \put(20,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdfLV}[3]{\GdfV{#3} \put(30,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GefLV}[3]{\GefV{#3} \put(40,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GffLV}[3]{\GffV{#3} \put(50,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GgfLV}[3]{\GgfV{#3} \put(60,50){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GagLV}[3]{\GagV{#3} \put(0,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GbgLV}[3]{\GbgV{#3} \put(10,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GcgLV}[3]{\GcgV{#3} \put(20,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GdgLV}[3]{\GdgV{#3} \put(30,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GegLV}[3]{\GegV{#3} \put(40,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GfgLV}[3]{\GfgV{#3} \put(50,60){\makebox(0,0){\hspace{#1}#2}}} \newcommand{\GggLV}[3]{\GggV{#3} \put(60,60){\makebox(0,0){\hspace{#1}#2}}} % % THE GRID - UNLABELED EDGES % \newcommand{\Gaaba}{\put(0,0){\line(1,0){10}}} \newcommand{\Gaaca}{\put(0,0){\line(1,0){20}}} \newcommand{\Gaada}{\put(0,0){\line(1,0){30}}} \newcommand{\Gaaea}{\put(0,0){\line(1,0){40}}} \newcommand{\Gaafa}{\put(0,0){\line(1,0){50}}} \newcommand{\Gaaga}{\put(0,0){\line(1,0){60}}} \newcommand{\Gaaab}{\put(0,0){\line(0,1){10}}} \newcommand{\Gaabb}{\put(0,0){\line(1,1){10}}} \newcommand{\Gaacb}{\put(0,0){\line(2,1){20}}} \newcommand{\Gaadb}{\put(0,0){\line(3,1){30}}} \newcommand{\Gaaeb}{\put(0,0){\line(4,1){40}}} \newcommand{\Gaafb}{\put(0,0){\line(5,1){50}}} \newcommand{\Gaagb}{\put(0,0){\line(6,1){60}}} \newcommand{\Gaaac}{\put(0,0){\line(0,1){20}}} \newcommand{\Gaabc}{\put(0,0){\line(1,2){10}}} \newcommand{\Gaacc}{\put(0,0){\line(1,1){20}}} \newcommand{\Gaadc}{\put(0,0){\line(3,2){30}}} \newcommand{\Gaaec}{\put(0,0){\line(2,1){40}}} \newcommand{\Gaafc}{\put(0,0){\line(5,2){50}}} \newcommand{\Gaagc}{\put(0,0){\line(3,1){60}}} \newcommand{\Gaaad}{\put(0,0){\line(0,1){30}}} \newcommand{\Gaabd}{\put(0,0){\line(1,3){10}}} \newcommand{\Gaacd}{\put(0,0){\line(2,3){20}}} \newcommand{\Gaadd}{\put(0,0){\line(1,1){30}}} \newcommand{\Gaaed}{\put(0,0){\line(4,3){40}}} \newcommand{\Gaafd}{\put(0,0){\line(5,3){50}}} \newcommand{\Gaagd}{\put(0,0){\line(2,1){60}}} \newcommand{\Gaaae}{\put(0,0){\line(0,1){40}}} \newcommand{\Gaabe}{\put(0,0){\line(1,4){10}}} \newcommand{\Gaace}{\put(0,0){\line(1,2){20}}} \newcommand{\Gaade}{\put(0,0){\line(3,4){30}}} \newcommand{\Gaaee}{\put(0,0){\line(1,1){40}}} \newcommand{\Gaafe}{\put(0,0){\line(5,4){50}}} \newcommand{\Gaage}{\put(0,0){\line(3,2){60}}} \newcommand{\Gaaaf}{\put(0,0){\line(0,1){50}}} \newcommand{\Gaabf}{\put(0,0){\line(1,5){10}}} \newcommand{\Gaacf}{\put(0,0){\line(2,5){20}}} \newcommand{\Gaadf}{\put(0,0){\line(3,5){30}}} \newcommand{\Gaaef}{\put(0,0){\line(4,5){40}}} \newcommand{\Gaaff}{\put(0,0){\line(1,1){50}}} \newcommand{\Gaagf}{\put(0,0){\line(6,5){60}}} \newcommand{\Gaaag}{\put(0,0){\line(0,1){60}}} \newcommand{\Gaabg}{\put(0,0){\line(1,6){10}}} \newcommand{\Gaacg}{\put(0,0){\line(1,3){20}}} \newcommand{\Gaadg}{\put(0,0){\line(1,2){30}}} \newcommand{\Gaaeg}{\put(0,0){\line(2,3){40}}} \newcommand{\Gaafg}{\put(0,0){\line(5,6){50}}} \newcommand{\Gaagg}{\put(0,0){\line(1,1){60}}} %---- \newcommand{\Gbaca}{\put(10,0){\line(1,0){10}}} \newcommand{\Gbada}{\put(10,0){\line(1,0){20}}} \newcommand{\Gbaea}{\put(10,0){\line(1,0){30}}} \newcommand{\Gbafa}{\put(10,0){\line(1,0){40}}} \newcommand{\Gbaga}{\put(10,0){\line(1,0){50}}} \newcommand{\Gbaab}{\put(10,0){\line(-1,1){10}}} \newcommand{\Gbabb}{\put(10,0){\line(0,1){10}}} \newcommand{\Gbacb}{\put(10,0){\line(1,1){10}}} \newcommand{\Gbadb}{\put(10,0){\line(2,1){20}}} \newcommand{\Gbaeb}{\put(10,0){\line(3,1){30}}} \newcommand{\Gbafb}{\put(10,0){\line(4,1){40}}} \newcommand{\Gbagb}{\put(10,0){\line(5,1){50}}} \newcommand{\Gbaac}{\put(10,0){\line(-1,2){10}}} \newcommand{\Gbabc}{\put(10,0){\line(0,1){20}}} \newcommand{\Gbacc}{\put(10,0){\line(1,2){10}}} \newcommand{\Gbadc}{\put(10,0){\line(1,1){20}}} \newcommand{\Gbaec}{\put(10,0){\line(3,2){30}}} \newcommand{\Gbafc}{\put(10,0){\line(2,1){40}}} \newcommand{\Gbagc}{\put(10,0){\line(5,2){50}}} \newcommand{\Gbaad}{\put(10,0){\line(-1,3){10}}} \newcommand{\Gbabd}{\put(10,0){\line(0,1){30}}} \newcommand{\Gbacd}{\put(10,0){\line(1,3){10}}} \newcommand{\Gbadd}{\put(10,0){\line(2,3){20}}} \newcommand{\Gbaed}{\put(10,0){\line(1,1){30}}} \newcommand{\Gbafd}{\put(10,0){\line(4,3){40}}} \newcommand{\Gbagd}{\put(10,0){\line(5,3){50}}} \newcommand{\Gbaae}{\put(10,0){\line(-1,4){10}}} \newcommand{\Gbabe}{\put(10,0){\line(0,1){40}}} \newcommand{\Gbace}{\put(10,0){\line(1,4){10}}} \newcommand{\Gbade}{\put(10,0){\line(1,2){20}}} \newcommand{\Gbaee}{\put(10,0){\line(3,4){30}}} \newcommand{\Gbafe}{\put(10,0){\line(1,1){40}}} \newcommand{\Gbage}{\put(10,0){\line(5,4){50}}} \newcommand{\Gbaaf}{\put(10,0){\line(-1,5){10}}} \newcommand{\Gbabf}{\put(10,0){\line(0,1){50}}} \newcommand{\Gbacf}{\put(10,0){\line(1,5){10}}} \newcommand{\Gbadf}{\put(10,0){\line(2,5){20}}} \newcommand{\Gbaef}{\put(10,0){\line(3,5){30}}} \newcommand{\Gbaff}{\put(10,0){\line(4,5){40}}} \newcommand{\Gbagf}{\put(10,0){\line(1,1){50}}} \newcommand{\Gbaag}{\put(10,0){\line(-1,6){10}}} \newcommand{\Gbabg}{\put(10,0){\line(0,1){60}}} \newcommand{\Gbacg}{\put(10,0){\line(1,6){10}}} \newcommand{\Gbadg}{\put(10,0){\line(1,3){20}}} \newcommand{\Gbaeg}{\put(10,0){\line(1,2){30}}} \newcommand{\Gbafg}{\put(10,0){\line(2,3){40}}} \newcommand{\Gbagg}{\put(10,0){\line(5,6){50}}} %---- \newcommand{\Gcada}{\put(20,0){\line(1,0){10}}} \newcommand{\Gcaea}{\put(20,0){\line(1,0){20}}} \newcommand{\Gcafa}{\put(20,0){\line(1,0){30}}} \newcommand{\Gcaga}{\put(20,0){\line(1,0){40}}} \newcommand{\Gcaab}{\put(20,0){\line(-2,1){20}}} \newcommand{\Gcabb}{\put(20,0){\line(-1,1){10}}} \newcommand{\Gcacb}{\put(20,0){\line(0,1){10}}} \newcommand{\Gcadb}{\put(20,0){\line(1,1){10}}} \newcommand{\Gcaeb}{\put(20,0){\line(2,1){20}}} \newcommand{\Gcafb}{\put(20,0){\line(3,1){30}}} \newcommand{\Gcagb}{\put(20,0){\line(4,1){40}}} \newcommand{\Gcaac}{\put(20,0){\line(-1,1){20}}} \newcommand{\Gcabc}{\put(20,0){\line(-1,2){10}}} \newcommand{\Gcacc}{\put(20,0){\line(0,1){20}}} \newcommand{\Gcadc}{\put(20,0){\line(1,2){10}}} \newcommand{\Gcaec}{\put(20,0){\line(1,1){20}}} \newcommand{\Gcafc}{\put(20,0){\line(3,2){30}}} \newcommand{\Gcagc}{\put(20,0){\line(2,1){40}}} \newcommand{\Gcaad}{\put(20,0){\line(-2,3){20}}} \newcommand{\Gcabd}{\put(20,0){\line(-1,3){10}}} \newcommand{\Gcacd}{\put(20,0){\line(0,1){30}}} \newcommand{\Gcadd}{\put(20,0){\line(1,3){10}}} \newcommand{\Gcaed}{\put(20,0){\line(2,3){20}}} \newcommand{\Gcafd}{\put(20,0){\line(1,1){30}}} \newcommand{\Gcagd}{\put(20,0){\line(4,3){40}}} \newcommand{\Gcaae}{\put(20,0){\line(-1,2){20}}} \newcommand{\Gcabe}{\put(20,0){\line(-1,4){10}}} \newcommand{\Gcace}{\put(20,0){\line(0,1){40}}} \newcommand{\Gcade}{\put(20,0){\line(1,4){10}}} \newcommand{\Gcaee}{\put(20,0){\line(1,2){20}}} \newcommand{\Gcafe}{\put(20,0){\line(3,4){30}}} \newcommand{\Gcage}{\put(20,0){\line(1,1){40}}} \newcommand{\Gcaaf}{\put(20,0){\line(-2,5){20}}} \newcommand{\Gcabf}{\put(20,0){\line(-1,5){10}}} \newcommand{\Gcacf}{\put(20,0){\line(0,1){50}}} \newcommand{\Gcadf}{\put(20,0){\line(1,5){10}}} \newcommand{\Gcaef}{\put(20,0){\line(2,5){20}}} \newcommand{\Gcaff}{\put(20,0){\line(3,5){30}}} \newcommand{\Gcagf}{\put(20,0){\line(4,5){40}}} \newcommand{\Gcaag}{\put(20,0){\line(-1,3){20}}} \newcommand{\Gcabg}{\put(20,0){\line(-1,6){10}}} \newcommand{\Gcacg}{\put(20,0){\line(0,1){60}}} \newcommand{\Gcadg}{\put(20,0){\line(1,6){10}}} \newcommand{\Gcaeg}{\put(20,0){\line(1,3){20}}} \newcommand{\Gcafg}{\put(20,0){\line(1,2){30}}} \newcommand{\Gcagg}{\put(20,0){\line(2,3){40}}} %------ \newcommand{\Gdaea}{\put(30,0){\line(1,0){10}}} \newcommand{\Gdafa}{\put(30,0){\line(1,0){20}}} \newcommand{\Gdaga}{\put(30,0){\line(1,0){30}}} \newcommand{\Gdaab}{\put(30,0){\line(-3,1){30}}} \newcommand{\Gdabb}{\put(30,0){\line(-2,1){20}}} \newcommand{\Gdacb}{\put(30,0){\line(-1,1){10}}} \newcommand{\Gdadb}{\put(30,0){\line(0,1){10}}} \newcommand{\Gdaeb}{\put(30,0){\line(1,1){10}}} \newcommand{\Gdafb}{\put(30,0){\line(2,1){20}}} \newcommand{\Gdagb}{\put(30,0){\line(3,1){30}}} \newcommand{\Gdaac}{\put(30,0){\line(-3,2){30}}} \newcommand{\Gdabc}{\put(30,0){\line(-1,1){20}}} \newcommand{\Gdacc}{\put(30,0){\line(-1,2){10}}} \newcommand{\Gdadc}{\put(30,0){\line(0,1){20}}} \newcommand{\Gdaec}{\put(30,0){\line(1,2){10}}} \newcommand{\Gdafc}{\put(30,0){\line(1,1){20}}} \newcommand{\Gdagc}{\put(30,0){\line(3,2){30}}} \newcommand{\Gdaad}{\put(30,0){\line(-1,1){30}}} \newcommand{\Gdabd}{\put(30,0){\line(-2,3){20}}} \newcommand{\Gdacd}{\put(30,0){\line(-1,3){10}}} \newcommand{\Gdadd}{\put(30,0){\line(0,1){30}}} \newcommand{\Gdaed}{\put(30,0){\line(1,3){10}}} \newcommand{\Gdafd}{\put(30,0){\line(2,3){20}}} \newcommand{\Gdagd}{\put(30,0){\line(1,1){30}}} \newcommand{\Gdaae}{\put(30,0){\line(-3,4){30}}} \newcommand{\Gdabe}{\put(30,0){\line(-1,2){20}}} \newcommand{\Gdace}{\put(30,0){\line(-1,4){10}}} \newcommand{\Gdade}{\put(30,0){\line(0,1){40}}} \newcommand{\Gdaee}{\put(30,0){\line(1,4){10}}} \newcommand{\Gdafe}{\put(30,0){\line(1,2){20}}} \newcommand{\Gdage}{\put(30,0){\line(3,4){30}}} \newcommand{\Gdaaf}{\put(30,0){\line(-3,5){30}}} \newcommand{\Gdabf}{\put(30,0){\line(-2,5){20}}} \newcommand{\Gdacf}{\put(30,0){\line(-1,5){10}}} \newcommand{\Gdadf}{\put(30,0){\line(0,1){50}}} \newcommand{\Gdaef}{\put(30,0){\line(1,5){10}}} \newcommand{\Gdaff}{\put(30,0){\line(2,5){20}}} \newcommand{\Gdagf}{\put(30,0){\line(3,5){30}}} \newcommand{\Gdaag}{\put(30,0){\line(-1,2){30}}} \newcommand{\Gdabg}{\put(30,0){\line(-1,3){20}}} \newcommand{\Gdacg}{\put(30,0){\line(-1,6){10}}} \newcommand{\Gdadg}{\put(30,0){\line(0,1){60}}} \newcommand{\Gdaeg}{\put(30,0){\line(1,6){10}}} \newcommand{\Gdafg}{\put(30,0){\line(1,3){20}}} \newcommand{\Gdagg}{\put(30,0){\line(1,2){30}}} %----- \newcommand{\Geafa}{\put(40,0){\line(1,0){10}}} \newcommand{\Geaga}{\put(40,0){\line(1,0){20}}} \newcommand{\Geaab}{\put(40,0){\line(-4,1){40}}} \newcommand{\Geabb}{\put(40,0){\line(-3,1){30}}} \newcommand{\Geacb}{\put(40,0){\line(-2,1){20}}} \newcommand{\Geadb}{\put(40,0){\line(-1,1){10}}} \newcommand{\Geaeb}{\put(40,0){\line(0,1){10}}} \newcommand{\Geafb}{\put(40,0){\line(1,1){10}}} \newcommand{\Geagb}{\put(40,0){\line(2,1){20}}} \newcommand{\Geaac}{\put(40,0){\line(-2,1){40}}} \newcommand{\Geabc}{\put(40,0){\line(-3,2){30}}} \newcommand{\Geacc}{\put(40,0){\line(-1,1){20}}} \newcommand{\Geadc}{\put(40,0){\line(-1,2){10}}} \newcommand{\Geaec}{\put(40,0){\line(0,1){20}}} \newcommand{\Geafc}{\put(40,0){\line(1,2){10}}} \newcommand{\Geagc}{\put(40,0){\line(1,1){20}}} \newcommand{\Geaad}{\put(40,0){\line(-4,3){40}}} \newcommand{\Geabd}{\put(40,0){\line(-1,1){30}}} \newcommand{\Geacd}{\put(40,0){\line(-2,3){20}}} \newcommand{\Geadd}{\put(40,0){\line(-1,3){10}}} \newcommand{\Geaed}{\put(40,0){\line(0,1){30}}} \newcommand{\Geafd}{\put(40,0){\line(1,3){10}}} \newcommand{\Geagd}{\put(40,0){\line(2,3){20}}} \newcommand{\Geaae}{\put(40,0){\line(-1,1){40}}} \newcommand{\Geabe}{\put(40,0){\line(-3,4){30}}} \newcommand{\Geace}{\put(40,0){\line(-1,2){20}}} \newcommand{\Geade}{\put(40,0){\line(-1,4){10}}} \newcommand{\Geaee}{\put(40,0){\line(0,1){40}}} \newcommand{\Geafe}{\put(40,0){\line(1,4){10}}} \newcommand{\Geage}{\put(40,0){\line(1,2){20}}} \newcommand{\Geaaf}{\put(40,0){\line(-4,5){40}}} \newcommand{\Geabf}{\put(40,0){\line(-3,5){30}}} \newcommand{\Geacf}{\put(40,0){\line(-2,5){20}}} \newcommand{\Geadf}{\put(40,0){\line(-1,5){10}}} \newcommand{\Geaef}{\put(40,0){\line(0,1){50}}} \newcommand{\Geaff}{\put(40,0){\line(1,5){10}}} \newcommand{\Geagf}{\put(40,0){\line(2,5){20}}} \newcommand{\Geaag}{\put(40,0){\line(-2,3){40}}} \newcommand{\Geabg}{\put(40,0){\line(-1,2){30}}} \newcommand{\Geacg}{\put(40,0){\line(-1,3){20}}} \newcommand{\Geadg}{\put(40,0){\line(-1,6){10}}} \newcommand{\Geaeg}{\put(40,0){\line(0,1){60}}} \newcommand{\Geafg}{\put(40,0){\line(1,6){10}}} \newcommand{\Geagg}{\put(40,0){\line(1,3){20}}} %----- \newcommand{\Gfaga}{\put(50,0){\line(1,0){10}}} \newcommand{\Gfaab}{\put(50,0){\line(-5,1){50}}} \newcommand{\Gfabb}{\put(50,0){\line(-4,1){40}}} \newcommand{\Gfacb}{\put(50,0){\line(-3,1){30}}} \newcommand{\Gfadb}{\put(50,0){\line(-2,1){20}}} \newcommand{\Gfaeb}{\put(50,0){\line(-1,0){10}}} \newcommand{\Gfafb}{\put(50,0){\line(0,1){10}}} \newcommand{\Gfagb}{\put(50,0){\line(1,1){10}}} \newcommand{\Gfaac}{\put(50,0){\line(-5,2){50}}} \newcommand{\Gfabc}{\put(50,0){\line(-2,1){40}}} \newcommand{\Gfacc}{\put(50,0){\line(-3,2){30}}} \newcommand{\Gfadc}{\put(50,0){\line(-1,1){20}}} \newcommand{\Gfaec}{\put(50,0){\line(-1,2){10}}} \newcommand{\Gfafc}{\put(50,0){\line(0,1){20}}} \newcommand{\Gfagc}{\put(50,0){\line(1,2){10}}} \newcommand{\Gfaad}{\put(50,0){\line(-5,3){50}}} \newcommand{\Gfabd}{\put(50,0){\line(-4,3){40}}} \newcommand{\Gfacd}{\put(50,0){\line(-1,1){30}}} \newcommand{\Gfadd}{\put(50,0){\line(-2,3){20}}} \newcommand{\Gfaed}{\put(50,0){\line(-1,3){10}}} \newcommand{\Gfafd}{\put(50,0){\line(0,1){30}}} \newcommand{\Gfagd}{\put(50,0){\line(1,3){10}}} \newcommand{\Gfaae}{\put(50,0){\line(-5,4){50}}} \newcommand{\Gfabe}{\put(50,0){\line(-1,1){40}}} \newcommand{\Gface}{\put(50,0){\line(-3,4){30}}} \newcommand{\Gfade}{\put(50,0){\line(-1,2){20}}} \newcommand{\Gfaee}{\put(50,0){\line(-1,4){10}}} \newcommand{\Gfafe}{\put(50,0){\line(0,1){40}}} \newcommand{\Gfage}{\put(50,0){\line(1,4){10}}} \newcommand{\Gfaaf}{\put(50,0){\line(-1,1){50}}} \newcommand{\Gfabf}{\put(50,0){\line(-4,5){40}}} \newcommand{\Gfacf}{\put(50,0){\line(-3,5){30}}} \newcommand{\Gfadf}{\put(50,0){\line(-2,5){20}}} \newcommand{\Gfaef}{\put(50,0){\line(-1,5){10}}} \newcommand{\Gfaff}{\put(50,0){\line(0,1){50}}} \newcommand{\Gfagf}{\put(50,0){\line(1,5){10}}} \newcommand{\Gfaag}{\put(50,0){\line(-5,6){50}}} \newcommand{\Gfabg}{\put(50,0){\line(-2,3){40}}} \newcommand{\Gfacg}{\put(50,0){\line(-1,2){30}}} \newcommand{\Gfadg}{\put(50,0){\line(-1,3){20}}} \newcommand{\Gfaeg}{\put(50,0){\line(-1,6){10}}} \newcommand{\Gfafg}{\put(50,0){\line(0,1){60}}} \newcommand{\Gfagg}{\put(50,0){\line(1,6){10}}} %---- \newcommand{\Ggaab}{\put(60,0){\line(-6,1){60}}} \newcommand{\Ggabb}{\put(60,0){\line(-5,1){50}}} \newcommand{\Ggacb}{\put(60,0){\line(-4,1){40}}} \newcommand{\Ggadb}{\put(60,0){\line(-3,1){30}}} \newcommand{\Ggaeb}{\put(60,0){\line(-2,0){20}}} \newcommand{\Ggafb}{\put(60,0){\line(-1,1){10}}} \newcommand{\Ggagb}{\put(60,0){\line(0,1){10}}} \newcommand{\Ggaac}{\put(60,0){\line(-3,1){60}}} \newcommand{\Ggabc}{\put(60,0){\line(-5,2){50}}} \newcommand{\Ggacc}{\put(60,0){\line(-2,1){40}}} \newcommand{\Ggadc}{\put(60,0){\line(-3,2){30}}} \newcommand{\Ggaec}{\put(60,0){\line(-1,1){20}}} \newcommand{\Ggafc}{\put(60,0){\line(-1,2){10}}} \newcommand{\Ggagc}{\put(60,0){\line(0,1){20}}} \newcommand{\Ggaad}{\put(60,0){\line(-2,1){60}}} \newcommand{\Ggabd}{\put(60,0){\line(-6,5){50}}} \newcommand{\Ggacd}{\put(60,0){\line(-3,2){40}}} \newcommand{\Ggadd}{\put(60,0){\line(-1,1){30}}} \newcommand{\Ggaed}{\put(60,0){\line(-2,3){20}}} \newcommand{\Ggafd}{\put(60,0){\line(-1,3){10}}} \newcommand{\Ggagd}{\put(60,0){\line(0,1){30}}} \newcommand{\Ggaae}{\put(60,0){\line(-3,2){60}}} \newcommand{\Ggabe}{\put(60,0){\line(-5,4){50}}} \newcommand{\Ggace}{\put(60,0){\line(-1,1){40}}} \newcommand{\Ggade}{\put(60,0){\line(-3,4){30}}} \newcommand{\Ggaee}{\put(60,0){\line(-1,2){20}}} \newcommand{\Ggafe}{\put(60,0){\line(-1,4){10}}} \newcommand{\Ggage}{\put(60,0){\line(0,1){40}}} \newcommand{\Ggaaf}{\put(60,0){\line(-6,5){60}}} \newcommand{\Ggabf}{\put(60,0){\line(-1,1){50}}} \newcommand{\Ggacf}{\put(60,0){\line(-4,5){40}}} \newcommand{\Ggadf}{\put(60,0){\line(-3,5){30}}} \newcommand{\Ggaef}{\put(60,0){\line(-3,5){20}}} \newcommand{\Ggaff}{\put(60,0){\line(-1,5){10}}} \newcommand{\Ggagf}{\put(60,0){\line(0,1){50}}} \newcommand{\Ggaag}{\put(60,0){\line(-1,1){60}}} \newcommand{\Ggabg}{\put(60,0){\line(-5,6){50}}} \newcommand{\Ggacg}{\put(60,0){\line(-2,3){40}}} \newcommand{\Ggadg}{\put(60,0){\line(-1,2){30}}} \newcommand{\Ggaeg}{\put(60,0){\line(-1,3){20}}} \newcommand{\Ggafg}{\put(60,0){\line(-1,6){10}}} \newcommand{\Ggagg}{\put(60,0){\line(0,1){60}}} %----- \newcommand{\Gabbb}{\put(0,10){\line(1,0){10}}} \newcommand{\Gabcb}{\put(0,10){\line(1,0){20}}} \newcommand{\Gabdb}{\put(0,10){\line(1,0){30}}} \newcommand{\Gabeb}{\put(0,10){\line(1,0){40}}} \newcommand{\Gabfb}{\put(0,10){\line(1,0){50}}} \newcommand{\Gabgb}{\put(0,10){\line(1,0){60}}} \newcommand{\Gabac}{\put(0,10){\line(0,1){10}}} \newcommand{\Gabbc}{\put(0,10){\line(1,1){10}}} \newcommand{\Gabcc}{\put(0,10){\line(2,1){20}}} \newcommand{\Gabdc}{\put(0,10){\line(3,1){30}}} \newcommand{\Gabec}{\put(0,10){\line(4,1){40}}} \newcommand{\Gabfc}{\put(0,10){\line(5,1){50}}} \newcommand{\Gabgc}{\put(0,10){\line(6,1){60}}} \newcommand{\Gabad}{\put(0,10){\line(0,1){20}}} \newcommand{\Gabbd}{\put(0,10){\line(1,2){10}}} \newcommand{\Gabcd}{\put(0,10){\line(1,1){20}}} \newcommand{\Gabdd}{\put(0,10){\line(3,2){30}}} \newcommand{\Gabed}{\put(0,10){\line(2,1){40}}} \newcommand{\Gabfd}{\put(0,10){\line(5,2){50}}} \newcommand{\Gabgd}{\put(0,10){\line(3,1){60}}} \newcommand{\Gabae}{\put(0,10){\line(0,1){30}}} \newcommand{\Gabbe}{\put(0,10){\line(1,3){10}}} \newcommand{\Gabce}{\put(0,10){\line(2,3){20}}} \newcommand{\Gabde}{\put(0,10){\line(1,1){30}}} \newcommand{\Gabee}{\put(0,10){\line(4,3){40}}} \newcommand{\Gabfe}{\put(0,10){\line(5,3){50}}} \newcommand{\Gabge}{\put(0,10){\line(2,1){60}}} \newcommand{\Gabaf}{\put(0,10){\line(0,1){40}}} \newcommand{\Gabbf}{\put(0,10){\line(1,4){10}}} \newcommand{\Gabcf}{\put(0,10){\line(1,2){20}}} \newcommand{\Gabdf}{\put(0,10){\line(3,4){30}}} \newcommand{\Gabef}{\put(0,10){\line(1,1){40}}} \newcommand{\Gabff}{\put(0,10){\line(5,4){50}}} \newcommand{\Gabgf}{\put(0,10){\line(3,2){60}}} \newcommand{\Gabag}{\put(0,10){\line(0,1){50}}} \newcommand{\Gabbg}{\put(0,10){\line(1,5){10}}} \newcommand{\Gabcg}{\put(0,10){\line(2,5){20}}} \newcommand{\Gabdg}{\put(0,10){\line(3,5){30}}} \newcommand{\Gabeg}{\put(0,10){\line(4,5){40}}} \newcommand{\Gabfg}{\put(0,10){\line(1,1){50}}} \newcommand{\Gabgg}{\put(0,10){\line(6,5){60}}} %---- \newcommand{\Gbbcb}{\put(10,10){\line(1,0){10}}} \newcommand{\Gbbdb}{\put(10,10){\line(1,0){20}}} \newcommand{\Gbbeb}{\put(10,10){\line(1,0){30}}} \newcommand{\Gbbfb}{\put(10,10){\line(1,0){40}}} \newcommand{\Gbbgb}{\put(10,10){\line(1,0){50}}} \newcommand{\Gbbac}{\put(10,10){\line(-1,1){10}}} \newcommand{\Gbbbc}{\put(10,10){\line(0,1){10}}} \newcommand{\Gbbcc}{\put(10,10){\line(1,1){10}}} \newcommand{\Gbbdc}{\put(10,10){\line(2,1){20}}} \newcommand{\Gbbec}{\put(10,10){\line(3,1){30}}} \newcommand{\Gbbfc}{\put(10,10){\line(4,1){40}}} \newcommand{\Gbbgc}{\put(10,10){\line(5,1){50}}} \newcommand{\Gbbad}{\put(10,10){\line(-1,2){10}}} \newcommand{\Gbbbd}{\put(10,10){\line(0,1){20}}} \newcommand{\Gbbcd}{\put(10,10){\line(1,2){10}}} \newcommand{\Gbbdd}{\put(10,10){\line(1,1){20}}} \newcommand{\Gbbed}{\put(10,10){\line(3,2){30}}} \newcommand{\Gbbfd}{\put(10,10){\line(2,1){40}}} \newcommand{\Gbbgd}{\put(10,10){\line(5,2){50}}} \newcommand{\Gbbae}{\put(10,10){\line(-1,3){10}}} \newcommand{\Gbbbe}{\put(10,10){\line(0,1){30}}} \newcommand{\Gbbce}{\put(10,10){\line(1,3){10}}} \newcommand{\Gbbde}{\put(10,10){\line(2,3){20}}} \newcommand{\Gbbee}{\put(10,10){\line(1,1){30}}} \newcommand{\Gbbfe}{\put(10,10){\line(4,3){40}}} \newcommand{\Gbbge}{\put(10,10){\line(5,3){50}}} \newcommand{\Gbbaf}{\put(10,10){\line(-1,4){10}}} \newcommand{\Gbbbf}{\put(10,10){\line(0,1){40}}} \newcommand{\Gbbcf}{\put(10,10){\line(1,4){10}}} \newcommand{\Gbbdf}{\put(10,10){\line(1,2){20}}} \newcommand{\Gbbef}{\put(10,10){\line(3,4){30}}} \newcommand{\Gbbff}{\put(10,10){\line(1,1){40}}} \newcommand{\Gbbgf}{\put(10,10){\line(5,4){50}}} \newcommand{\Gbbag}{\put(10,10){\line(-1,5){10}}} \newcommand{\Gbbbg}{\put(10,10){\line(0,1){50}}} \newcommand{\Gbbcg}{\put(10,10){\line(1,5){10}}} \newcommand{\Gbbdg}{\put(10,10){\line(2,5){20}}} \newcommand{\Gbbeg}{\put(10,10){\line(3,5){30}}} \newcommand{\Gbbfg}{\put(10,10){\line(4,5){40}}} \newcommand{\Gbbgg}{\put(10,10){\line(1,1){50}}} %---- \newcommand{\Gcbdb}{\put(20,10){\line(1,0){10}}} \newcommand{\Gcbeb}{\put(20,10){\line(1,0){20}}} \newcommand{\Gcbfb}{\put(20,10){\line(1,0){30}}} \newcommand{\Gcbgb}{\put(20,10){\line(1,0){40}}} \newcommand{\Gcbac}{\put(20,10){\line(-2,1){20}}} \newcommand{\Gcbbc}{\put(20,10){\line(-1,1){10}}} \newcommand{\Gcbcc}{\put(20,10){\line(0,1){10}}} \newcommand{\Gcbdc}{\put(20,10){\line(1,1){10}}} \newcommand{\Gcbec}{\put(20,10){\line(2,1){20}}} \newcommand{\Gcbfc}{\put(20,10){\line(3,1){30}}} \newcommand{\Gcbgc}{\put(20,10){\line(4,1){40}}} \newcommand{\Gcbad}{\put(20,10){\line(-1,1){20}}} \newcommand{\Gcbbd}{\put(20,10){\line(-1,2){10}}} \newcommand{\Gcbcd}{\put(20,10){\line(0,1){20}}} \newcommand{\Gcbdd}{\put(20,10){\line(1,2){10}}} \newcommand{\Gcbed}{\put(20,10){\line(1,1){20}}} \newcommand{\Gcbfd}{\put(20,10){\line(3,2){30}}} \newcommand{\Gcbgd}{\put(20,10){\line(2,1){40}}} \newcommand{\Gcbae}{\put(20,10){\line(-2,3){20}}} \newcommand{\Gcbbe}{\put(20,10){\line(-1,3){10}}} \newcommand{\Gcbce}{\put(20,10){\line(0,1){30}}} \newcommand{\Gcbde}{\put(20,10){\line(1,3){10}}} \newcommand{\Gcbee}{\put(20,10){\line(2,3){20}}} \newcommand{\Gcbfe}{\put(20,10){\line(1,1){30}}} \newcommand{\Gcbge}{\put(20,10){\line(4,3){40}}} \newcommand{\Gcbaf}{\put(20,10){\line(-1,2){20}}} \newcommand{\Gcbbf}{\put(20,10){\line(-1,4){10}}} \newcommand{\Gcbcf}{\put(20,10){\line(0,1){40}}} \newcommand{\Gcbdf}{\put(20,10){\line(1,4){10}}} \newcommand{\Gcbef}{\put(20,10){\line(1,2){20}}} \newcommand{\Gcbff}{\put(20,10){\line(3,4){30}}} \newcommand{\Gcbgf}{\put(20,10){\line(1,1){40}}} \newcommand{\Gcbag}{\put(20,10){\line(-2,5){20}}} \newcommand{\Gcbbg}{\put(20,10){\line(-1,5){10}}} \newcommand{\Gcbcg}{\put(20,10){\line(0,1){50}}} \newcommand{\Gcbdg}{\put(20,10){\line(1,5){10}}} \newcommand{\Gcbeg}{\put(20,10){\line(2,5){20}}} \newcommand{\Gcbfg}{\put(20,10){\line(3,5){30}}} \newcommand{\Gcbgg}{\put(20,10){\line(4,5){40}}} %------ \newcommand{\Gdbeb}{\put(30,10){\line(1,0){10}}} \newcommand{\Gdbfb}{\put(30,10){\line(1,0){20}}} \newcommand{\Gdbgb}{\put(30,10){\line(1,0){30}}} \newcommand{\Gdbac}{\put(30,10){\line(-3,1){30}}} \newcommand{\Gdbbc}{\put(30,10){\line(-2,1){20}}} \newcommand{\Gdbcc}{\put(30,10){\line(-1,1){10}}} \newcommand{\Gdbdc}{\put(30,10){\line(0,1){10}}} \newcommand{\Gdbec}{\put(30,10){\line(1,1){10}}} \newcommand{\Gdbfc}{\put(30,10){\line(2,1){20}}} \newcommand{\Gdbgc}{\put(30,10){\line(3,1){30}}} \newcommand{\Gdbad}{\put(30,10){\line(-3,2){30}}} \newcommand{\Gdbbd}{\put(30,10){\line(-1,1){20}}} \newcommand{\Gdbcd}{\put(30,10){\line(-1,2){10}}} \newcommand{\Gdbdd}{\put(30,10){\line(0,1){20}}} \newcommand{\Gdbed}{\put(30,10){\line(1,2){10}}} \newcommand{\Gdbfd}{\put(30,10){\line(1,1){20}}} \newcommand{\Gdbgd}{\put(30,10){\line(3,2){30}}} \newcommand{\Gdbae}{\put(30,10){\line(-1,1){30}}} \newcommand{\Gdbbe}{\put(30,10){\line(-2,3){20}}} \newcommand{\Gdbce}{\put(30,10){\line(-1,3){10}}} \newcommand{\Gdbde}{\put(30,10){\line(0,1){30}}} \newcommand{\Gdbee}{\put(30,10){\line(1,3){10}}} \newcommand{\Gdbfe}{\put(30,10){\line(2,3){20}}} \newcommand{\Gdbge}{\put(30,10){\line(1,1){30}}} \newcommand{\Gdbaf}{\put(30,10){\line(-3,4){30}}} \newcommand{\Gdbbf}{\put(30,10){\line(-1,2){20}}} \newcommand{\Gdbcf}{\put(30,10){\line(-1,4){10}}} \newcommand{\Gdbdf}{\put(30,10){\line(0,1){40}}} \newcommand{\Gdbef}{\put(30,10){\line(1,4){10}}} \newcommand{\Gdbff}{\put(30,10){\line(1,2){20}}} \newcommand{\Gdbgf}{\put(30,10){\line(3,4){30}}} \newcommand{\Gdbag}{\put(30,10){\line(-3,5){30}}} \newcommand{\Gdbbg}{\put(30,10){\line(-2,5){20}}} \newcommand{\Gdbcg}{\put(30,10){\line(-1,5){10}}} \newcommand{\Gdbdg}{\put(30,10){\line(0,1){50}}} \newcommand{\Gdbeg}{\put(30,10){\line(1,5){10}}} \newcommand{\Gdbfg}{\put(30,10){\line(2,5){20}}} \newcommand{\Gdbgg}{\put(30,10){\line(3,5){30}}} %----- \newcommand{\Gebfb}{\put(40,10){\line(1,0){10}}} \newcommand{\Gebgb}{\put(40,10){\line(1,0){20}}} \newcommand{\Gebac}{\put(40,10){\line(-4,1){40}}} \newcommand{\Gebbc}{\put(40,10){\line(-3,1){30}}} \newcommand{\Gebcc}{\put(40,10){\line(-2,1){20}}} \newcommand{\Gebdc}{\put(40,10){\line(-1,1){10}}} \newcommand{\Gebec}{\put(40,10){\line(0,1){10}}} \newcommand{\Gebfc}{\put(40,10){\line(1,1){10}}} \newcommand{\Gebgc}{\put(40,10){\line(2,1){20}}} \newcommand{\Gebad}{\put(40,10){\line(-2,1){40}}} \newcommand{\Gebbd}{\put(40,10){\line(-3,2){30}}} \newcommand{\Gebcd}{\put(40,10){\line(-1,1){20}}} \newcommand{\Gebdd}{\put(40,10){\line(-1,2){10}}} \newcommand{\Gebed}{\put(40,10){\line(0,1){20}}} \newcommand{\Gebfd}{\put(40,10){\line(1,2){10}}} \newcommand{\Gebgd}{\put(40,10){\line(1,1){20}}} \newcommand{\Gebae}{\put(40,10){\line(-4,3){40}}} \newcommand{\Gebbe}{\put(40,10){\line(-1,1){30}}} \newcommand{\Gebce}{\put(40,10){\line(-2,3){20}}} \newcommand{\Gebde}{\put(40,10){\line(-1,3){10}}} \newcommand{\Gebee}{\put(40,10){\line(0,1){30}}} \newcommand{\Gebfe}{\put(40,10){\line(1,3){10}}} \newcommand{\Gebge}{\put(40,10){\line(2,3){20}}} \newcommand{\Gebaf}{\put(40,10){\line(-1,1){40}}} \newcommand{\Gebbf}{\put(40,10){\line(-3,4){30}}} \newcommand{\Gebcf}{\put(40,10){\line(-1,2){20}}} \newcommand{\Gebdf}{\put(40,10){\line(-1,4){10}}} \newcommand{\Gebef}{\put(40,10){\line(0,1){40}}} \newcommand{\Gebff}{\put(40,10){\line(1,4){10}}} \newcommand{\Gebgf}{\put(40,10){\line(1,2){20}}} \newcommand{\Gebag}{\put(40,10){\line(-4,5){40}}} \newcommand{\Gebbg}{\put(40,10){\line(-3,5){30}}} \newcommand{\Gebcg}{\put(40,10){\line(-2,5){20}}} \newcommand{\Gebdg}{\put(40,10){\line(-1,5){10}}} \newcommand{\Gebeg}{\put(40,10){\line(0,1){50}}} \newcommand{\Gebfg}{\put(40,10){\line(1,5){10}}} \newcommand{\Gebgg}{\put(40,10){\line(2,5){20}}} %----- \newcommand{\Gfbgb}{\put(50,10){\line(1,0){10}}} \newcommand{\Gfbac}{\put(50,10){\line(-5,1){50}}} \newcommand{\Gfbbc}{\put(50,10){\line(-4,1){40}}} \newcommand{\Gfbcc}{\put(50,10){\line(-3,1){30}}} \newcommand{\Gfbdc}{\put(50,10){\line(-2,1){20}}} \newcommand{\Gfbec}{\put(50,10){\line(-1,0){10}}} \newcommand{\Gfbfc}{\put(50,10){\line(0,1){10}}} \newcommand{\Gfbgc}{\put(50,10){\line(1,1){10}}} \newcommand{\Gfbad}{\put(50,10){\line(-5,2){50}}} \newcommand{\Gfbbd}{\put(50,10){\line(-2,1){40}}} \newcommand{\Gfbcd}{\put(50,10){\line(-3,2){30}}} \newcommand{\Gfbdd}{\put(50,10){\line(-1,1){20}}} \newcommand{\Gfbed}{\put(50,10){\line(-1,2){10}}} \newcommand{\Gfbfd}{\put(50,10){\line(0,1){20}}} \newcommand{\Gfbgd}{\put(50,10){\line(1,2){10}}} \newcommand{\Gfbae}{\put(50,10){\line(-5,3){50}}} \newcommand{\Gfbbe}{\put(50,10){\line(-4,3){40}}} \newcommand{\Gfbce}{\put(50,10){\line(-1,1){30}}} \newcommand{\Gfbde}{\put(50,10){\line(-2,3){20}}} \newcommand{\Gfbee}{\put(50,10){\line(-1,3){10}}} \newcommand{\Gfbfe}{\put(50,10){\line(0,1){30}}} \newcommand{\Gfbge}{\put(50,10){\line(1,3){10}}} \newcommand{\Gfbaf}{\put(50,10){\line(-5,4){50}}} \newcommand{\Gfbbf}{\put(50,10){\line(-1,1){40}}} \newcommand{\Gfbcf}{\put(50,10){\line(-3,4){30}}} \newcommand{\Gfbdf}{\put(50,10){\line(-1,2){20}}} \newcommand{\Gfbef}{\put(50,10){\line(-1,4){10}}} \newcommand{\Gfbff}{\put(50,10){\line(0,1){40}}} \newcommand{\Gfbgf}{\put(50,10){\line(1,4){10}}} \newcommand{\Gfbag}{\put(50,10){\line(-1,1){50}}} \newcommand{\Gfbbg}{\put(50,10){\line(-4,5){40}}} \newcommand{\Gfbcg}{\put(50,10){\line(-3,5){30}}} \newcommand{\Gfbdg}{\put(50,10){\line(-2,5){20}}} \newcommand{\Gfbeg}{\put(50,10){\line(-1,5){10}}} \newcommand{\Gfbfg}{\put(50,10){\line(0,1){50}}} \newcommand{\Gfbgg}{\put(50,10){\line(1,5){10}}} %---- \newcommand{\Ggbac}{\put(60,10){\line(-6,1){60}}} \newcommand{\Ggbbc}{\put(60,10){\line(-5,1){50}}} \newcommand{\Ggbcc}{\put(60,10){\line(-4,1){40}}} \newcommand{\Ggbdc}{\put(60,10){\line(-3,1){30}}} \newcommand{\Ggbec}{\put(60,10){\line(-2,0){20}}} \newcommand{\Ggbfc}{\put(60,10){\line(-1,1){10}}} \newcommand{\Ggbgc}{\put(60,10){\line(0,1){10}}} \newcommand{\Ggbad}{\put(60,10){\line(-3,1){60}}} \newcommand{\Ggbbd}{\put(60,10){\line(-5,2){50}}} \newcommand{\Ggbcd}{\put(60,10){\line(-2,1){40}}} \newcommand{\Ggbdd}{\put(60,10){\line(-3,2){30}}} \newcommand{\Ggbed}{\put(60,10){\line(-1,1){20}}} \newcommand{\Ggbfd}{\put(60,10){\line(-1,2){10}}} \newcommand{\Ggbgd}{\put(60,10){\line(0,1){20}}} \newcommand{\Ggbae}{\put(60,10){\line(-2,1){60}}} \newcommand{\Ggbbe}{\put(60,10){\line(-6,5){50}}} \newcommand{\Ggbce}{\put(60,10){\line(-3,2){40}}} \newcommand{\Ggbde}{\put(60,10){\line(-1,1){30}}} \newcommand{\Ggbee}{\put(60,10){\line(-2,3){20}}} \newcommand{\Ggbfe}{\put(60,10){\line(-1,3){10}}} \newcommand{\Ggbge}{\put(60,10){\line(0,1){30}}} \newcommand{\Ggbaf}{\put(60,10){\line(-3,2){60}}} \newcommand{\Ggbbf}{\put(60,10){\line(-5,4){50}}} \newcommand{\Ggbcf}{\put(60,10){\line(-1,1){40}}} \newcommand{\Ggbdf}{\put(60,10){\line(-3,4){30}}} \newcommand{\Ggbef}{\put(60,10){\line(-1,2){20}}} \newcommand{\Ggbff}{\put(60,10){\line(-1,4){10}}} \newcommand{\Ggbgf}{\put(60,10){\line(0,1){40}}} \newcommand{\Ggbag}{\put(60,10){\line(-6,5){60}}} \newcommand{\Ggbbg}{\put(60,10){\line(-1,1){50}}} \newcommand{\Ggbcg}{\put(60,10){\line(-4,5){40}}} \newcommand{\Ggbdg}{\put(60,10){\line(-3,5){30}}} \newcommand{\Ggbeg}{\put(60,10){\line(-3,5){20}}} \newcommand{\Ggbfg}{\put(60,10){\line(-1,5){10}}} \newcommand{\Ggbgg}{\put(60,10){\line(0,1){50}}} %----- \newcommand{\Gacbc}{\put(0,20){\line(1,0){10}}} \newcommand{\Gaccc}{\put(0,20){\line(1,0){20}}} \newcommand{\Gacdc}{\put(0,20){\line(1,0){30}}} \newcommand{\Gacec}{\put(0,20){\line(1,0){40}}} \newcommand{\Gacfc}{\put(0,20){\line(1,0){50}}} \newcommand{\Gacgc}{\put(0,20){\line(1,0){60}}} \newcommand{\Gacad}{\put(0,20){\line(0,1){10}}} \newcommand{\Gacbd}{\put(0,20){\line(1,1){10}}} \newcommand{\Gaccd}{\put(0,20){\line(2,1){20}}} \newcommand{\Gacdd}{\put(0,20){\line(3,1){30}}} \newcommand{\Gaced}{\put(0,20){\line(4,1){40}}} \newcommand{\Gacfd}{\put(0,20){\line(5,1){50}}} \newcommand{\Gacgd}{\put(0,20){\line(6,1){60}}} \newcommand{\Gacae}{\put(0,20){\line(0,1){20}}} \newcommand{\Gacbe}{\put(0,20){\line(1,2){10}}} \newcommand{\Gacce}{\put(0,20){\line(1,1){20}}} \newcommand{\Gacde}{\put(0,20){\line(3,2){30}}} \newcommand{\Gacee}{\put(0,20){\line(2,1){40}}} \newcommand{\Gacfe}{\put(0,20){\line(5,2){50}}} \newcommand{\Gacge}{\put(0,20){\line(3,1){60}}} \newcommand{\Gacaf}{\put(0,20){\line(0,1){30}}} \newcommand{\Gacbf}{\put(0,20){\line(1,3){10}}} \newcommand{\Gaccf}{\put(0,20){\line(2,3){20}}} \newcommand{\Gacdf}{\put(0,20){\line(1,1){30}}} \newcommand{\Gacef}{\put(0,20){\line(4,3){40}}} \newcommand{\Gacff}{\put(0,20){\line(5,3){50}}} \newcommand{\Gacgf}{\put(0,20){\line(2,1){60}}} \newcommand{\Gacag}{\put(0,20){\line(0,1){40}}} \newcommand{\Gacbg}{\put(0,20){\line(1,4){10}}} \newcommand{\Gaccg}{\put(0,20){\line(1,2){20}}} \newcommand{\Gacdg}{\put(0,20){\line(3,4){30}}} \newcommand{\Gaceg}{\put(0,20){\line(1,1){40}}} \newcommand{\Gacfg}{\put(0,20){\line(5,4){50}}} \newcommand{\Gacgg}{\put(0,20){\line(3,2){60}}} %---- \newcommand{\Gbccc}{\put(10,20){\line(1,0){10}}} \newcommand{\Gbcdc}{\put(10,20){\line(1,0){20}}} \newcommand{\Gbcec}{\put(10,20){\line(1,0){30}}} \newcommand{\Gbcfc}{\put(10,20){\line(1,0){40}}} \newcommand{\Gbcgc}{\put(10,20){\line(1,0){50}}} \newcommand{\Gbcad}{\put(10,20){\line(-1,1){10}}} \newcommand{\Gbcbd}{\put(10,20){\line(0,1){10}}} \newcommand{\Gbccd}{\put(10,20){\line(1,1){10}}} \newcommand{\Gbcdd}{\put(10,20){\line(2,1){20}}} \newcommand{\Gbced}{\put(10,20){\line(3,1){30}}} \newcommand{\Gbcfd}{\put(10,20){\line(4,1){40}}} \newcommand{\Gbcgd}{\put(10,20){\line(5,1){50}}} \newcommand{\Gbcae}{\put(10,20){\line(-1,2){10}}} \newcommand{\Gbcbe}{\put(10,20){\line(0,1){20}}} \newcommand{\Gbcce}{\put(10,20){\line(1,2){10}}} \newcommand{\Gbcde}{\put(10,20){\line(1,1){20}}} \newcommand{\Gbcee}{\put(10,20){\line(3,2){30}}} \newcommand{\Gbcfe}{\put(10,20){\line(2,1){40}}} \newcommand{\Gbcge}{\put(10,20){\line(5,2){50}}} \newcommand{\Gbcaf}{\put(10,20){\line(-1,3){10}}} \newcommand{\Gbcbf}{\put(10,20){\line(0,1){30}}} \newcommand{\Gbccf}{\put(10,20){\line(1,3){10}}} \newcommand{\Gbcdf}{\put(10,20){\line(2,3){20}}} \newcommand{\Gbcef}{\put(10,20){\line(1,1){30}}} \newcommand{\Gbcff}{\put(10,20){\line(4,3){40}}} \newcommand{\Gbcgf}{\put(10,20){\line(5,3){50}}} \newcommand{\Gbcag}{\put(10,20){\line(-1,4){10}}} \newcommand{\Gbcbg}{\put(10,20){\line(0,1){40}}} \newcommand{\Gbccg}{\put(10,20){\line(1,4){10}}} \newcommand{\Gbcdg}{\put(10,20){\line(1,2){20}}} \newcommand{\Gbceg}{\put(10,20){\line(3,4){30}}} \newcommand{\Gbcfg}{\put(10,20){\line(1,1){40}}} \newcommand{\Gbcgg}{\put(10,20){\line(5,4){50}}} %---- \newcommand{\Gccdc}{\put(20,20){\line(1,0){10}}} \newcommand{\Gccec}{\put(20,20){\line(1,0){20}}} \newcommand{\Gccfc}{\put(20,20){\line(1,0){30}}} \newcommand{\Gccgc}{\put(20,20){\line(1,0){40}}} \newcommand{\Gccad}{\put(20,20){\line(-2,1){20}}} \newcommand{\Gccbd}{\put(20,20){\line(-1,1){10}}} \newcommand{\Gcccd}{\put(20,20){\line(0,1){10}}} \newcommand{\Gccdd}{\put(20,20){\line(1,1){10}}} \newcommand{\Gcced}{\put(20,20){\line(2,1){20}}} \newcommand{\Gccfd}{\put(20,20){\line(3,1){30}}} \newcommand{\Gccgd}{\put(20,20){\line(4,1){40}}} \newcommand{\Gccae}{\put(20,20){\line(-1,1){20}}} \newcommand{\Gccbe}{\put(20,20){\line(-1,2){10}}} \newcommand{\Gccce}{\put(20,20){\line(0,1){20}}} \newcommand{\Gccde}{\put(20,20){\line(1,2){10}}} \newcommand{\Gccee}{\put(20,20){\line(1,1){20}}} \newcommand{\Gccfe}{\put(20,20){\line(3,2){30}}} \newcommand{\Gccge}{\put(20,20){\line(2,1){40}}} \newcommand{\Gccaf}{\put(20,20){\line(-2,3){20}}} \newcommand{\Gccbf}{\put(20,20){\line(-1,3){10}}} \newcommand{\Gcccf}{\put(20,20){\line(0,1){30}}} \newcommand{\Gccdf}{\put(20,20){\line(1,3){10}}} \newcommand{\Gccef}{\put(20,20){\line(2,3){20}}} \newcommand{\Gccff}{\put(20,20){\line(1,1){30}}} \newcommand{\Gccgf}{\put(20,20){\line(4,3){40}}} \newcommand{\Gccag}{\put(20,20){\line(-1,2){20}}} \newcommand{\Gccbg}{\put(20,20){\line(-1,4){10}}} \newcommand{\Gcccg}{\put(20,20){\line(0,1){40}}} \newcommand{\Gccdg}{\put(20,20){\line(1,4){10}}} \newcommand{\Gcceg}{\put(20,20){\line(1,2){20}}} \newcommand{\Gccfg}{\put(20,20){\line(3,4){30}}} \newcommand{\Gccgg}{\put(20,20){\line(1,1){40}}} %------ \newcommand{\Gdcec}{\put(30,20){\line(1,0){10}}} \newcommand{\Gdcfc}{\put(30,20){\line(1,0){20}}} \newcommand{\Gdcgc}{\put(30,20){\line(1,0){30}}} \newcommand{\Gdcad}{\put(30,20){\line(-3,1){30}}} \newcommand{\Gdcbd}{\put(30,20){\line(-2,1){20}}} \newcommand{\Gdccd}{\put(30,20){\line(-1,1){10}}} \newcommand{\Gdcdd}{\put(30,20){\line(0,1){10}}} \newcommand{\Gdced}{\put(30,20){\line(1,1){10}}} \newcommand{\Gdcfd}{\put(30,20){\line(2,1){20}}} \newcommand{\Gdcgd}{\put(30,20){\line(3,1){30}}} \newcommand{\Gdcae}{\put(30,20){\line(-3,2){30}}} \newcommand{\Gdcbe}{\put(30,20){\line(-1,1){20}}} \newcommand{\Gdcce}{\put(30,20){\line(-1,2){10}}} \newcommand{\Gdcde}{\put(30,20){\line(0,1){20}}} \newcommand{\Gdcee}{\put(30,20){\line(1,2){10}}} \newcommand{\Gdcfe}{\put(30,20){\line(1,1){20}}} \newcommand{\Gdcge}{\put(30,20){\line(3,2){30}}} \newcommand{\Gdcaf}{\put(30,20){\line(-1,1){30}}} \newcommand{\Gdcbf}{\put(30,20){\line(-2,3){20}}} \newcommand{\Gdccf}{\put(30,20){\line(-1,3){10}}} \newcommand{\Gdcdf}{\put(30,20){\line(0,1){30}}} \newcommand{\Gdcef}{\put(30,20){\line(1,3){10}}} \newcommand{\Gdcff}{\put(30,20){\line(2,3){20}}} \newcommand{\Gdcgf}{\put(30,20){\line(1,1){30}}} \newcommand{\Gdcag}{\put(30,20){\line(-3,4){30}}} \newcommand{\Gdcbg}{\put(30,20){\line(-1,2){20}}} \newcommand{\Gdccg}{\put(30,20){\line(-1,4){10}}} \newcommand{\Gdcdg}{\put(30,20){\line(0,1){40}}} \newcommand{\Gdceg}{\put(30,20){\line(1,4){10}}} \newcommand{\Gdcfg}{\put(30,20){\line(1,2){20}}} \newcommand{\Gdcgg}{\put(30,20){\line(3,4){30}}} %----- \newcommand{\Gecfc}{\put(40,20){\line(1,0){10}}} \newcommand{\Gecgc}{\put(40,20){\line(1,0){20}}} \newcommand{\Gecad}{\put(40,20){\line(-4,1){40}}} \newcommand{\Gecbd}{\put(40,20){\line(-3,1){30}}} \newcommand{\Geccd}{\put(40,20){\line(-2,1){20}}} \newcommand{\Gecdd}{\put(40,20){\line(-1,1){10}}} \newcommand{\Geced}{\put(40,20){\line(0,1){10}}} \newcommand{\Gecfd}{\put(40,20){\line(1,1){10}}} \newcommand{\Gecgd}{\put(40,20){\line(2,1){20}}} \newcommand{\Gecae}{\put(40,20){\line(-2,1){40}}} \newcommand{\Gecbe}{\put(40,20){\line(-3,2){30}}} \newcommand{\Gecce}{\put(40,20){\line(-1,1){20}}} \newcommand{\Gecde}{\put(40,20){\line(-1,2){10}}} \newcommand{\Gecee}{\put(40,20){\line(0,1){20}}} \newcommand{\Gecfe}{\put(40,20){\line(1,2){10}}} \newcommand{\Gecge}{\put(40,20){\line(1,1){20}}} \newcommand{\Gecaf}{\put(40,20){\line(-4,3){40}}} \newcommand{\Gecbf}{\put(40,20){\line(-1,1){30}}} \newcommand{\Geccf}{\put(40,20){\line(-2,3){20}}} \newcommand{\Gecdf}{\put(40,20){\line(-1,3){10}}} \newcommand{\Gecef}{\put(40,20){\line(0,1){30}}} \newcommand{\Gecff}{\put(40,20){\line(1,3){10}}} \newcommand{\Gecgf}{\put(40,20){\line(2,3){20}}} \newcommand{\Gecag}{\put(40,20){\line(-1,1){40}}} \newcommand{\Gecbg}{\put(40,20){\line(-3,4){30}}} \newcommand{\Geccg}{\put(40,20){\line(-1,2){20}}} \newcommand{\Gecdg}{\put(40,20){\line(-1,4){10}}} \newcommand{\Geceg}{\put(40,20){\line(0,1){40}}} \newcommand{\Gecfg}{\put(40,20){\line(1,4){10}}} \newcommand{\Gecgg}{\put(40,20){\line(1,2){20}}} %----- \newcommand{\Gfcgc}{\put(50,20){\line(1,0){10}}} \newcommand{\Gfcad}{\put(50,20){\line(-5,1){50}}} \newcommand{\Gfcbd}{\put(50,20){\line(-4,1){40}}} \newcommand{\Gfccd}{\put(50,20){\line(-3,1){30}}} \newcommand{\Gfcdd}{\put(50,20){\line(-2,1){20}}} \newcommand{\Gfced}{\put(50,20){\line(-1,0){10}}} \newcommand{\Gfcfd}{\put(50,20){\line(0,1){10}}} \newcommand{\Gfcgd}{\put(50,20){\line(1,1){10}}} \newcommand{\Gfcae}{\put(50,20){\line(-5,2){50}}} \newcommand{\Gfcbe}{\put(50,20){\line(-2,1){40}}} \newcommand{\Gfcce}{\put(50,20){\line(-3,2){30}}} \newcommand{\Gfcde}{\put(50,20){\line(-1,1){20}}} \newcommand{\Gfcee}{\put(50,20){\line(-1,2){10}}} \newcommand{\Gfcfe}{\put(50,20){\line(0,1){20}}} \newcommand{\Gfcge}{\put(50,20){\line(1,2){10}}} \newcommand{\Gfcaf}{\put(50,20){\line(-5,3){50}}} \newcommand{\Gfcbf}{\put(50,20){\line(-4,3){40}}} \newcommand{\Gfccf}{\put(50,20){\line(-1,1){30}}} \newcommand{\Gfcdf}{\put(50,20){\line(-2,3){20}}} \newcommand{\Gfcef}{\put(50,20){\line(-1,3){10}}} \newcommand{\Gfcff}{\put(50,20){\line(0,1){30}}} \newcommand{\Gfcgf}{\put(50,20){\line(1,3){10}}} \newcommand{\Gfcag}{\put(50,20){\line(-5,4){50}}} \newcommand{\Gfcbg}{\put(50,20){\line(-1,1){40}}} \newcommand{\Gfccg}{\put(50,20){\line(-3,4){30}}} \newcommand{\Gfcdg}{\put(50,20){\line(-1,2){20}}} \newcommand{\Gfceg}{\put(50,20){\line(-1,4){10}}} \newcommand{\Gfcfg}{\put(50,20){\line(0,1){40}}} \newcommand{\Gfcgg}{\put(50,20){\line(1,4){10}}} %---- \newcommand{\Ggcad}{\put(60,20){\line(-6,1){60}}} \newcommand{\Ggcbd}{\put(60,20){\line(-5,1){50}}} \newcommand{\Ggccd}{\put(60,20){\line(-4,1){40}}} \newcommand{\Ggcdd}{\put(60,20){\line(-3,1){30}}} \newcommand{\Ggced}{\put(60,20){\line(-2,0){20}}} \newcommand{\Ggcfd}{\put(60,20){\line(-1,1){10}}} \newcommand{\Ggcgd}{\put(60,20){\line(0,1){10}}} \newcommand{\Ggcae}{\put(60,20){\line(-3,1){60}}} \newcommand{\Ggcbe}{\put(60,20){\line(-5,2){50}}} \newcommand{\Ggcce}{\put(60,20){\line(-2,1){40}}} \newcommand{\Ggcde}{\put(60,20){\line(-3,2){30}}} \newcommand{\Ggcee}{\put(60,20){\line(-1,1){20}}} \newcommand{\Ggcfe}{\put(60,20){\line(-1,2){10}}} \newcommand{\Ggcge}{\put(60,20){\line(0,1){20}}} \newcommand{\Ggcaf}{\put(60,20){\line(-2,1){60}}} \newcommand{\Ggcbf}{\put(60,20){\line(-6,5){50}}} \newcommand{\Ggccf}{\put(60,20){\line(-3,2){40}}} \newcommand{\Ggcdf}{\put(60,20){\line(-1,1){30}}} \newcommand{\Ggcef}{\put(60,20){\line(-2,3){20}}} \newcommand{\Ggcff}{\put(60,20){\line(-1,3){10}}} \newcommand{\Ggcgf}{\put(60,20){\line(0,1){30}}} \newcommand{\Ggcag}{\put(60,20){\line(-3,2){60}}} \newcommand{\Ggcbg}{\put(60,20){\line(-5,4){50}}} \newcommand{\Ggccg}{\put(60,20){\line(-1,1){40}}} \newcommand{\Ggcdg}{\put(60,20){\line(-3,4){30}}} \newcommand{\Ggceg}{\put(60,20){\line(-1,2){20}}} \newcommand{\Ggcfg}{\put(60,20){\line(-1,4){10}}} \newcommand{\Ggcgg}{\put(60,20){\line(0,1){40}}} %----- \newcommand{\Gadbd}{\put(0,30){\line(1,0){10}}} \newcommand{\Gadcd}{\put(0,30){\line(1,0){20}}} \newcommand{\Gaddd}{\put(0,30){\line(1,0){30}}} \newcommand{\Gaded}{\put(0,30){\line(1,0){40}}} \newcommand{\Gadfd}{\put(0,30){\line(1,0){50}}} \newcommand{\Gadgd}{\put(0,30){\line(1,0){60}}} \newcommand{\Gadae}{\put(0,30){\line(0,1){10}}} \newcommand{\Gadbe}{\put(0,30){\line(1,1){10}}} \newcommand{\Gadce}{\put(0,30){\line(2,1){20}}} \newcommand{\Gadde}{\put(0,30){\line(3,1){30}}} \newcommand{\Gadee}{\put(0,30){\line(4,1){40}}} \newcommand{\Gadfe}{\put(0,30){\line(5,1){50}}} \newcommand{\Gadge}{\put(0,30){\line(6,1){60}}} \newcommand{\Gadaf}{\put(0,30){\line(0,1){20}}} \newcommand{\Gadbf}{\put(0,30){\line(1,2){10}}} \newcommand{\Gadcf}{\put(0,30){\line(1,1){20}}} \newcommand{\Gaddf}{\put(0,30){\line(3,2){30}}} \newcommand{\Gadef}{\put(0,30){\line(2,1){40}}} \newcommand{\Gadff}{\put(0,30){\line(5,2){50}}} \newcommand{\Gadgf}{\put(0,30){\line(3,1){60}}} \newcommand{\Gadag}{\put(0,30){\line(0,1){30}}} \newcommand{\Gadbg}{\put(0,30){\line(1,3){10}}} \newcommand{\Gadcg}{\put(0,30){\line(2,3){20}}} \newcommand{\Gaddg}{\put(0,30){\line(1,1){30}}} \newcommand{\Gadeg}{\put(0,30){\line(4,3){40}}} \newcommand{\Gadfg}{\put(0,30){\line(5,3){50}}} \newcommand{\Gadgg}{\put(0,30){\line(2,1){60}}} %---- \newcommand{\Gbdcd}{\put(10,30){\line(1,0){10}}} \newcommand{\Gbddd}{\put(10,30){\line(1,0){20}}} \newcommand{\Gbded}{\put(10,30){\line(1,0){30}}} \newcommand{\Gbdfd}{\put(10,30){\line(1,0){40}}} \newcommand{\Gbdgd}{\put(10,30){\line(1,0){50}}} \newcommand{\Gbdae}{\put(10,30){\line(-1,1){10}}} \newcommand{\Gbdbe}{\put(10,30){\line(0,1){10}}} \newcommand{\Gbdce}{\put(10,30){\line(1,1){10}}} \newcommand{\Gbdde}{\put(10,30){\line(2,1){20}}} \newcommand{\Gbdee}{\put(10,30){\line(3,1){30}}} \newcommand{\Gbdfe}{\put(10,30){\line(4,1){40}}} \newcommand{\Gbdge}{\put(10,30){\line(5,1){50}}} \newcommand{\Gbdaf}{\put(10,30){\line(-1,2){10}}} \newcommand{\Gbdbf}{\put(10,30){\line(0,1){20}}} \newcommand{\Gbdcf}{\put(10,30){\line(1,2){10}}} \newcommand{\Gbddf}{\put(10,30){\line(1,1){20}}} \newcommand{\Gbdef}{\put(10,30){\line(3,2){30}}} \newcommand{\Gbdff}{\put(10,30){\line(2,1){40}}} \newcommand{\Gbdgf}{\put(10,30){\line(5,2){50}}} \newcommand{\Gbdag}{\put(10,30){\line(-1,3){10}}} \newcommand{\Gbdbg}{\put(10,30){\line(0,1){30}}} \newcommand{\Gbdcg}{\put(10,30){\line(1,3){10}}} \newcommand{\Gbddg}{\put(10,30){\line(2,3){20}}} \newcommand{\Gbdeg}{\put(10,30){\line(1,1){30}}} \newcommand{\Gbdfg}{\put(10,30){\line(4,3){40}}} \newcommand{\Gbdgg}{\put(10,30){\line(5,3){50}}} %------ \newcommand{\Gcddd}{\put(20,30){\line(1,0){10}}} \newcommand{\Gcded}{\put(20,30){\line(1,0){20}}} \newcommand{\Gcdfd}{\put(20,30){\line(1,0){30}}} \newcommand{\Gcdgd}{\put(20,30){\line(1,0){40}}} \newcommand{\Gcdae}{\put(20,30){\line(-2,1){20}}} \newcommand{\Gcdbe}{\put(20,30){\line(-1,1){10}}} \newcommand{\Gcdce}{\put(20,30){\line(0,1){10}}} \newcommand{\Gcdde}{\put(20,30){\line(1,1){10}}} \newcommand{\Gcdee}{\put(20,30){\line(2,1){20}}} \newcommand{\Gcdfe}{\put(20,30){\line(3,1){30}}} \newcommand{\Gcdge}{\put(20,30){\line(4,1){40}}} \newcommand{\Gcdaf}{\put(20,30){\line(-1,1){20}}} \newcommand{\Gcdbf}{\put(20,30){\line(-1,2){10}}} \newcommand{\Gcdcf}{\put(20,30){\line(0,1){20}}} \newcommand{\Gcddf}{\put(20,30){\line(1,2){10}}} \newcommand{\Gcdef}{\put(20,30){\line(1,1){20}}} \newcommand{\Gcdff}{\put(20,30){\line(3,2){30}}} \newcommand{\Gcdgf}{\put(20,30){\line(2,1){40}}} \newcommand{\Gcdag}{\put(20,30){\line(-2,3){20}}} \newcommand{\Gcdbg}{\put(20,30){\line(-1,3){10}}} \newcommand{\Gcdcg}{\put(20,30){\line(0,1){30}}} \newcommand{\Gcddg}{\put(20,30){\line(1,3){10}}} \newcommand{\Gcdeg}{\put(20,30){\line(2,3){20}}} \newcommand{\Gcdfg}{\put(20,30){\line(1,1){30}}} \newcommand{\Gcdgg}{\put(20,30){\line(4,3){40}}} %------ \newcommand{\Gdded}{\put(30,30){\line(1,0){10}}} \newcommand{\Gddfd}{\put(30,30){\line(1,0){20}}} \newcommand{\Gddgd}{\put(30,30){\line(1,0){30}}} \newcommand{\Gddae}{\put(30,30){\line(-3,1){30}}} \newcommand{\Gddbe}{\put(30,30){\line(-2,1){20}}} \newcommand{\Gddce}{\put(30,30){\line(-1,1){10}}} \newcommand{\Gddde}{\put(30,30){\line(0,1){10}}} \newcommand{\Gddee}{\put(30,30){\line(1,1){10}}} \newcommand{\Gddfe}{\put(30,30){\line(2,1){20}}} \newcommand{\Gddge}{\put(30,30){\line(3,1){30}}} \newcommand{\Gddaf}{\put(30,30){\line(-3,2){30}}} \newcommand{\Gddbf}{\put(30,30){\line(-1,1){20}}} \newcommand{\Gddcf}{\put(30,30){\line(-1,2){10}}} \newcommand{\Gdddf}{\put(30,30){\line(0,1){20}}} \newcommand{\Gddef}{\put(30,30){\line(1,2){10}}} \newcommand{\Gddff}{\put(30,30){\line(1,1){20}}} \newcommand{\Gddgf}{\put(30,30){\line(3,2){30}}} \newcommand{\Gddag}{\put(30,30){\line(-1,1){30}}} \newcommand{\Gddbg}{\put(30,30){\line(-2,3){20}}} \newcommand{\Gddcg}{\put(30,30){\line(-1,3){10}}} \newcommand{\Gdddg}{\put(30,30){\line(0,1){30}}} \newcommand{\Gddeg}{\put(30,30){\line(1,3){10}}} \newcommand{\Gddfg}{\put(30,30){\line(2,3){20}}} \newcommand{\Gddgg}{\put(30,30){\line(1,1){30}}} %----- \newcommand{\Gedfd}{\put(40,30){\line(1,0){10}}} \newcommand{\Gedgd}{\put(40,30){\line(1,0){20}}} \newcommand{\Gedae}{\put(40,30){\line(-4,1){40}}} \newcommand{\Gedbe}{\put(40,30){\line(-3,1){30}}} \newcommand{\Gedce}{\put(40,30){\line(-2,1){20}}} \newcommand{\Gedde}{\put(40,30){\line(-1,1){10}}} \newcommand{\Gedee}{\put(40,30){\line(0,1){10}}} \newcommand{\Gedfe}{\put(40,30){\line(1,1){10}}} \newcommand{\Gedge}{\put(40,30){\line(2,1){20}}} \newcommand{\Gedaf}{\put(40,30){\line(-2,1){40}}} \newcommand{\Gedbf}{\put(40,30){\line(-3,2){30}}} \newcommand{\Gedcf}{\put(40,30){\line(-1,1){20}}} \newcommand{\Geddf}{\put(40,30){\line(-1,2){10}}} \newcommand{\Gedef}{\put(40,30){\line(0,1){20}}} \newcommand{\Gedff}{\put(40,30){\line(1,2){10}}} \newcommand{\Gedgf}{\put(40,30){\line(1,1){20}}} \newcommand{\Gedag}{\put(40,30){\line(-4,3){40}}} \newcommand{\Gedbg}{\put(40,30){\line(-1,1){30}}} \newcommand{\Gedcg}{\put(40,30){\line(-2,3){20}}} \newcommand{\Geddg}{\put(40,30){\line(-1,3){10}}} \newcommand{\Gedeg}{\put(40,30){\line(0,1){30}}} \newcommand{\Gedfg}{\put(40,30){\line(1,3){10}}} \newcommand{\Gedgg}{\put(40,30){\line(2,3){20}}} %----- \newcommand{\Gfdgd}{\put(50,30){\line(1,0){10}}} \newcommand{\Gfdae}{\put(50,30){\line(-5,1){50}}} \newcommand{\Gfdbe}{\put(50,30){\line(-4,1){40}}} \newcommand{\Gfdce}{\put(50,30){\line(-3,1){30}}} \newcommand{\Gfdde}{\put(50,30){\line(-2,1){20}}} \newcommand{\Gfdee}{\put(50,30){\line(-1,0){10}}} \newcommand{\Gfdfe}{\put(50,30){\line(0,1){10}}} \newcommand{\Gfdge}{\put(50,30){\line(1,1){10}}} \newcommand{\Gfdaf}{\put(50,30){\line(-5,2){50}}} \newcommand{\Gfdbf}{\put(50,30){\line(-2,1){40}}} \newcommand{\Gfdcf}{\put(50,30){\line(-3,2){30}}} \newcommand{\Gfddf}{\put(50,30){\line(-1,1){20}}} \newcommand{\Gfdef}{\put(50,30){\line(-1,2){10}}} \newcommand{\Gfdff}{\put(50,30){\line(0,1){20}}} \newcommand{\Gfdgf}{\put(50,30){\line(1,2){10}}} \newcommand{\Gfdag}{\put(50,30){\line(-5,3){50}}} \newcommand{\Gfdbg}{\put(50,30){\line(-4,3){40}}} \newcommand{\Gfdcg}{\put(50,30){\line(-1,1){30}}} \newcommand{\Gfddg}{\put(50,30){\line(-2,3){20}}} \newcommand{\Gfdeg}{\put(50,30){\line(-1,3){10}}} \newcommand{\Gfdfg}{\put(50,30){\line(0,1){30}}} \newcommand{\Gfdgg}{\put(50,30){\line(1,3){10}}} %---- \newcommand{\Ggdae}{\put(60,30){\line(-6,1){60}}} \newcommand{\Ggdbe}{\put(60,30){\line(-5,1){50}}} \newcommand{\Ggdce}{\put(60,30){\line(-4,1){40}}} \newcommand{\Ggdde}{\put(60,30){\line(-3,1){30}}} \newcommand{\Ggdee}{\put(60,30){\line(-2,0){20}}} \newcommand{\Ggdfe}{\put(60,30){\line(-1,1){10}}} \newcommand{\Ggdge}{\put(60,30){\line(0,1){10}}} \newcommand{\Ggdaf}{\put(60,30){\line(-3,1){60}}} \newcommand{\Ggdbf}{\put(60,30){\line(-5,2){50}}} \newcommand{\Ggdcf}{\put(60,30){\line(-2,1){40}}} \newcommand{\Ggddf}{\put(60,30){\line(-3,2){30}}} \newcommand{\Ggdef}{\put(60,30){\line(-1,1){20}}} \newcommand{\Ggdff}{\put(60,30){\line(-1,2){10}}} \newcommand{\Ggdgf}{\put(60,30){\line(0,1){20}}} \newcommand{\Ggdag}{\put(60,30){\line(-2,1){60}}} \newcommand{\Ggdbg}{\put(60,30){\line(-6,5){50}}} \newcommand{\Ggdcg}{\put(60,30){\line(-3,2){40}}} \newcommand{\Ggddg}{\put(60,30){\line(-1,1){30}}} \newcommand{\Ggdeg}{\put(60,30){\line(-2,3){20}}} \newcommand{\Ggdfg}{\put(60,30){\line(-1,3){10}}} \newcommand{\Ggdgg}{\put(60,30){\line(0,1){30}}} %----- \newcommand{\Gaebe}{\put(0,40){\line(1,0){10}}} \newcommand{\Gaece}{\put(0,40){\line(1,0){20}}} \newcommand{\Gaede}{\put(0,40){\line(1,0){30}}} \newcommand{\Gaeee}{\put(0,40){\line(1,0){40}}} \newcommand{\Gaefe}{\put(0,40){\line(1,0){50}}} \newcommand{\Gaege}{\put(0,40){\line(1,0){60}}} \newcommand{\Gaeaf}{\put(0,40){\line(0,1){10}}} \newcommand{\Gaebf}{\put(0,40){\line(1,1){10}}} \newcommand{\Gaecf}{\put(0,40){\line(2,1){20}}} \newcommand{\Gaedf}{\put(0,40){\line(3,1){30}}} \newcommand{\Gaeef}{\put(0,40){\line(4,1){40}}} \newcommand{\Gaeff}{\put(0,40){\line(5,1){50}}} \newcommand{\Gaegf}{\put(0,40){\line(6,1){60}}} \newcommand{\Gaeag}{\put(0,40){\line(0,1){20}}} \newcommand{\Gaebg}{\put(0,40){\line(1,2){10}}} \newcommand{\Gaecg}{\put(0,40){\line(1,1){20}}} \newcommand{\Gaedg}{\put(0,40){\line(3,2){30}}} \newcommand{\Gaeeg}{\put(0,40){\line(2,1){40}}} \newcommand{\Gaefg}{\put(0,40){\line(5,2){50}}} \newcommand{\Gaegg}{\put(0,40){\line(3,1){60}}} %---- \newcommand{\Gbece}{\put(10,40){\line(1,0){10}}} \newcommand{\Gbede}{\put(10,40){\line(1,0){20}}} \newcommand{\Gbeee}{\put(10,40){\line(1,0){30}}} \newcommand{\Gbefe}{\put(10,40){\line(1,0){40}}} \newcommand{\Gbege}{\put(10,40){\line(1,0){50}}} \newcommand{\Gbeaf}{\put(10,40){\line(-1,1){10}}} \newcommand{\Gbebf}{\put(10,40){\line(0,1){10}}} \newcommand{\Gbecf}{\put(10,40){\line(1,1){10}}} \newcommand{\Gbedf}{\put(10,40){\line(2,1){20}}} \newcommand{\Gbeef}{\put(10,40){\line(3,1){30}}} \newcommand{\Gbeff}{\put(10,40){\line(4,1){40}}} \newcommand{\Gbegf}{\put(10,40){\line(5,1){50}}} \newcommand{\Gbeag}{\put(10,40){\line(-1,2){10}}} \newcommand{\Gbebg}{\put(10,40){\line(0,1){20}}} \newcommand{\Gbecg}{\put(10,40){\line(1,2){10}}} \newcommand{\Gbedg}{\put(10,40){\line(1,1){20}}} \newcommand{\Gbeeg}{\put(10,40){\line(3,2){30}}} \newcommand{\Gbefg}{\put(10,40){\line(2,1){40}}} \newcommand{\Gbegg}{\put(10,40){\line(5,2){50}}} %------ \newcommand{\Gcede}{\put(20,40){\line(1,0){10}}} \newcommand{\Gceee}{\put(20,40){\line(1,0){20}}} \newcommand{\Gcefe}{\put(20,40){\line(1,0){30}}} \newcommand{\Gcege}{\put(20,40){\line(1,0){40}}} \newcommand{\Gceaf}{\put(20,40){\line(-2,1){20}}} \newcommand{\Gcebf}{\put(20,40){\line(-1,1){10}}} \newcommand{\Gcecf}{\put(20,40){\line(0,1){10}}} \newcommand{\Gcedf}{\put(20,40){\line(1,1){10}}} \newcommand{\Gceef}{\put(20,40){\line(2,1){20}}} \newcommand{\Gceff}{\put(20,40){\line(3,1){30}}} \newcommand{\Gcegf}{\put(20,40){\line(4,1){40}}} \newcommand{\Gceag}{\put(20,40){\line(-1,1){20}}} \newcommand{\Gcebg}{\put(20,40){\line(-1,2){10}}} \newcommand{\Gcecg}{\put(20,40){\line(0,1){20}}} \newcommand{\Gcedg}{\put(20,40){\line(1,2){10}}} \newcommand{\Gceeg}{\put(20,40){\line(1,1){20}}} \newcommand{\Gcefg}{\put(20,40){\line(3,2){30}}} \newcommand{\Gcegg}{\put(20,40){\line(2,1){40}}} %------ \newcommand{\Gdeee}{\put(30,40){\line(1,0){10}}} \newcommand{\Gdefe}{\put(30,40){\line(1,0){20}}} \newcommand{\Gdege}{\put(30,40){\line(1,0){30}}} \newcommand{\Gdeaf}{\put(30,40){\line(-3,1){30}}} \newcommand{\Gdebf}{\put(30,40){\line(-2,1){20}}} \newcommand{\Gdecf}{\put(30,40){\line(-1,1){10}}} \newcommand{\Gdedf}{\put(30,40){\line(0,1){10}}} \newcommand{\Gdeef}{\put(30,40){\line(1,1){10}}} \newcommand{\Gdeff}{\put(30,40){\line(2,1){20}}} \newcommand{\Gdegf}{\put(30,40){\line(3,1){30}}} \newcommand{\Gdeag}{\put(30,40){\line(-3,2){30}}} \newcommand{\Gdebg}{\put(30,40){\line(-1,1){20}}} \newcommand{\Gdecg}{\put(30,40){\line(-1,2){10}}} \newcommand{\Gdedg}{\put(30,40){\line(0,1){20}}} \newcommand{\Gdeeg}{\put(30,40){\line(1,2){10}}} \newcommand{\Gdefg}{\put(30,40){\line(1,1){20}}} \newcommand{\Gdegg}{\put(30,40){\line(3,2){30}}} %----- \newcommand{\Geefe}{\put(40,40){\line(1,0){10}}} \newcommand{\Geege}{\put(40,40){\line(1,0){20}}} \newcommand{\Geeaf}{\put(40,40){\line(-4,1){40}}} \newcommand{\Geebf}{\put(40,40){\line(-3,1){30}}} \newcommand{\Geecf}{\put(40,40){\line(-2,1){20}}} \newcommand{\Geedf}{\put(40,40){\line(-1,1){10}}} \newcommand{\Geeef}{\put(40,40){\line(0,1){10}}} \newcommand{\Geeff}{\put(40,40){\line(1,1){10}}} \newcommand{\Geegf}{\put(40,40){\line(2,1){20}}} \newcommand{\Geeag}{\put(40,40){\line(-2,1){40}}} \newcommand{\Geebg}{\put(40,40){\line(-3,2){30}}} \newcommand{\Geecg}{\put(40,40){\line(-1,1){20}}} \newcommand{\Geedg}{\put(40,40){\line(-1,2){10}}} \newcommand{\Geeeg}{\put(40,40){\line(0,1){20}}} \newcommand{\Geefg}{\put(40,40){\line(1,2){10}}} \newcommand{\Geegg}{\put(40,40){\line(1,1){20}}} %----- \newcommand{\Gfege}{\put(50,40){\line(1,0){10}}} \newcommand{\Gfeaf}{\put(50,40){\line(-5,1){50}}} \newcommand{\Gfebf}{\put(50,40){\line(-4,1){40}}} \newcommand{\Gfecf}{\put(50,40){\line(-3,1){30}}} \newcommand{\Gfedf}{\put(50,40){\line(-2,1){20}}} \newcommand{\Gfeef}{\put(50,40){\line(-1,0){10}}} \newcommand{\Gfeff}{\put(50,40){\line(0,1){10}}} \newcommand{\Gfegf}{\put(50,40){\line(1,1){10}}} \newcommand{\Gfeag}{\put(50,40){\line(-5,2){50}}} \newcommand{\Gfebg}{\put(50,40){\line(-2,1){40}}} \newcommand{\Gfecg}{\put(50,40){\line(-3,2){30}}} \newcommand{\Gfedg}{\put(50,40){\line(-1,1){20}}} \newcommand{\Gfeeg}{\put(50,40){\line(-1,2){10}}} \newcommand{\Gfefg}{\put(50,40){\line(0,1){20}}} \newcommand{\Gfegg}{\put(50,40){\line(1,2){10}}} %---- \newcommand{\Ggeaf}{\put(60,40){\line(-6,1){60}}} \newcommand{\Ggebf}{\put(60,40){\line(-5,1){50}}} \newcommand{\Ggecf}{\put(60,40){\line(-4,1){40}}} \newcommand{\Ggedf}{\put(60,40){\line(-3,1){30}}} \newcommand{\Ggeef}{\put(60,40){\line(-2,0){20}}} \newcommand{\Ggeff}{\put(60,40){\line(-1,1){10}}} \newcommand{\Ggegf}{\put(60,40){\line(0,1){10}}} \newcommand{\Ggeag}{\put(60,40){\line(-3,1){60}}} \newcommand{\Ggebg}{\put(60,40){\line(-5,2){50}}} \newcommand{\Ggecg}{\put(60,40){\line(-2,1){40}}} \newcommand{\Ggedg}{\put(60,40){\line(-3,2){30}}} \newcommand{\Ggeeg}{\put(60,40){\line(-1,1){20}}} \newcommand{\Ggefg}{\put(60,40){\line(-1,2){10}}} \newcommand{\Ggegg}{\put(60,40){\line(0,1){20}}} %----- \newcommand{\Gafbf}{\put(0,50){\line(1,0){10}}} \newcommand{\Gafcf}{\put(0,50){\line(1,0){20}}} \newcommand{\Gafdf}{\put(0,50){\line(1,0){30}}} \newcommand{\Gafef}{\put(0,50){\line(1,0){40}}} \newcommand{\Gafff}{\put(0,50){\line(1,0){50}}} \newcommand{\Gafgf}{\put(0,50){\line(1,0){60}}} \newcommand{\Gafag}{\put(0,50){\line(0,1){10}}} \newcommand{\Gafbg}{\put(0,50){\line(1,1){10}}} \newcommand{\Gafcg}{\put(0,50){\line(2,1){20}}} \newcommand{\Gafdg}{\put(0,50){\line(3,1){30}}} \newcommand{\Gafeg}{\put(0,50){\line(4,1){40}}} \newcommand{\Gaffg}{\put(0,50){\line(5,1){50}}} \newcommand{\Gafgg}{\put(0,50){\line(6,1){60}}} %---- \newcommand{\Gbfcf}{\put(10,50){\line(1,0){10}}} \newcommand{\Gbfdf}{\put(10,50){\line(1,0){20}}} \newcommand{\Gbfef}{\put(10,50){\line(1,0){30}}} \newcommand{\Gbfff}{\put(10,50){\line(1,0){40}}} \newcommand{\Gbfgf}{\put(10,50){\line(1,0){50}}} \newcommand{\Gbfag}{\put(10,50){\line(-1,1){10}}} \newcommand{\Gbfbg}{\put(10,50){\line(0,1){10}}} \newcommand{\Gbfcg}{\put(10,50){\line(1,1){10}}} \newcommand{\Gbfdg}{\put(10,50){\line(2,1){20}}} \newcommand{\Gbfeg}{\put(10,50){\line(3,1){30}}} \newcommand{\Gbffg}{\put(10,50){\line(4,1){40}}} \newcommand{\Gbfgg}{\put(10,50){\line(5,1){50}}} %------ \newcommand{\Gcfdf}{\put(20,50){\line(1,0){10}}} \newcommand{\Gcfef}{\put(20,50){\line(1,0){20}}} \newcommand{\Gcfff}{\put(20,50){\line(1,0){30}}} \newcommand{\Gcfgf}{\put(20,50){\line(1,0){40}}} \newcommand{\Gcfag}{\put(20,50){\line(-2,1){20}}} \newcommand{\Gcfbg}{\put(20,50){\line(-1,1){10}}} \newcommand{\Gcfcg}{\put(20,50){\line(0,1){10}}} \newcommand{\Gcfdg}{\put(20,50){\line(1,1){10}}} \newcommand{\Gcfeg}{\put(20,50){\line(2,1){20}}} \newcommand{\Gcffg}{\put(20,50){\line(3,1){30}}} \newcommand{\Gcfgg}{\put(20,50){\line(4,1){40}}} %------ \newcommand{\Gdfef}{\put(30,50){\line(1,0){10}}} \newcommand{\Gdfff}{\put(30,50){\line(1,0){20}}} \newcommand{\Gdfgf}{\put(30,50){\line(1,0){30}}} \newcommand{\Gdfag}{\put(30,50){\line(-3,1){30}}} \newcommand{\Gdfbg}{\put(30,50){\line(-2,1){20}}} \newcommand{\Gdfcg}{\put(30,50){\line(-1,1){10}}} \newcommand{\Gdfdg}{\put(30,50){\line(0,1){10}}} \newcommand{\Gdfeg}{\put(30,50){\line(1,1){10}}} \newcommand{\Gdffg}{\put(30,50){\line(2,1){20}}} \newcommand{\Gdfgg}{\put(30,50){\line(3,1){30}}} %----- \newcommand{\Gefff}{\put(40,50){\line(1,0){10}}} \newcommand{\Gefgf}{\put(40,50){\line(1,0){20}}} \newcommand{\Gefag}{\put(40,50){\line(-4,1){40}}} \newcommand{\Gefbg}{\put(40,50){\line(-3,1){30}}} \newcommand{\Gefcg}{\put(40,50){\line(-2,1){20}}} \newcommand{\Gefdg}{\put(40,50){\line(-1,1){10}}} \newcommand{\Gefeg}{\put(40,50){\line(0,1){10}}} \newcommand{\Geffg}{\put(40,50){\line(1,1){10}}} \newcommand{\Gefgg}{\put(40,50){\line(2,1){20}}} %----- \newcommand{\Gffgf}{\put(50,50){\line(1,0){10}}} \newcommand{\Gffag}{\put(50,50){\line(-5,1){50}}} \newcommand{\Gffbg}{\put(50,50){\line(-4,1){40}}} \newcommand{\Gffcg}{\put(50,50){\line(-3,1){30}}} \newcommand{\Gffdg}{\put(50,50){\line(-2,1){20}}} \newcommand{\Gffeg}{\put(50,50){\line(-1,0){10}}} \newcommand{\Gfffg}{\put(50,50){\line(0,1){10}}} \newcommand{\Gffgg}{\put(50,50){\line(1,1){10}}} %---- \newcommand{\Ggfag}{\put(60,50){\line(-6,1){60}}} \newcommand{\Ggfbg}{\put(60,50){\line(-5,1){50}}} \newcommand{\Ggfcg}{\put(60,50){\line(-4,1){40}}} \newcommand{\Ggfdg}{\put(60,50){\line(-3,1){30}}} \newcommand{\Ggfeg}{\put(60,50){\line(-2,0){20}}} \newcommand{\Ggffg}{\put(60,50){\line(-1,1){10}}} \newcommand{\Ggfgg}{\put(60,50){\line(0,1){10}}} %----- \newcommand{\Gagbg}{\put(0,60){\line(1,0){10}}} \newcommand{\Gagcg}{\put(0,60){\line(1,0){20}}} \newcommand{\Gagdg}{\put(0,60){\line(1,0){30}}} \newcommand{\Gageg}{\put(0,60){\line(1,0){40}}} \newcommand{\Gagfg}{\put(0,60){\line(1,0){50}}} \newcommand{\Gaggg}{\put(0,60){\line(1,0){60}}} %---- \newcommand{\Gbgcg}{\put(10,60){\line(1,0){10}}} \newcommand{\Gbgdg}{\put(10,60){\line(1,0){20}}} \newcommand{\Gbgeg}{\put(10,60){\line(1,0){30}}} \newcommand{\Gbgfg}{\put(10,60){\line(1,0){40}}} \newcommand{\Gbggg}{\put(10,60){\line(1,0){50}}} %---- \newcommand{\Gcgdg}{\put(20,60){\line(1,0){10}}} \newcommand{\Gcgeg}{\put(20,60){\line(1,0){20}}} \newcommand{\Gcgfg}{\put(20,60){\line(1,0){30}}} \newcommand{\Gcggg}{\put(20,60){\line(1,0){40}}} %---- \newcommand{\Gdgeg}{\put(30,60){\line(1,0){10}}} \newcommand{\Gdgfg}{\put(30,60){\line(1,0){20}}} \newcommand{\Gdggg}{\put(30,60){\line(1,0){30}}} %----- \newcommand{\Gegfg}{\put(40,60){\line(1,0){10}}} \newcommand{\Geggg}{\put(40,60){\line(1,0){20}}} %----- \newcommand{\Gfggg}{\put(50,60){\line(1,0){10}}} %----- \begin{document} \pagestyle{empty} \title{Why the characteristic polynomial factors } %\author{Bruce E. Sagan} %\address{Department of Mathematics\\ %Michigan State University\\ %East Lansing, MI 48824-1027} %\email{sagan@math.msu.edu} \author{Bruce E. Sagan\\ Department of Mathematics\\ Michigan State University\\ East Lansing, MI 48824-1027\\ sagan@math.msu.edu} %\keywords{characteristic polynomial, free arrangement, %M\"obius function, partially ordered set, signed graph, subspace %arrangement, supersolvable lattice} %\subjclass{Primary 06A07; Secondary 05C15, 20F55, 06C10, 52B30} \date{September 24, 1995 \\[1in] \begin{flushleft} Key Words: characteristic polynomial, free arrangement, M\"obius function, partially ordered set, signed graph, subspace arrangement, supersolvable lattice\\[1em] AMS subject classification (1991): Primary 06A07; Secondary 05C15, 20F55, 06C10, 52B30. \end{flushleft} } \maketitle \begin{flushleft} Proposed running head: \end{flushleft} \begin{center} Why $\chi$ factors \end{center} Send proofs to: \begin{center} Bruce E. Sagan \\ Department of Mathematics \\Michigan State University \\ East Lansing, MI 48824-1027\\[5pt] Tel.: 517-355-8329\\ FAX: 517-336-1562\\ Email: sagan@mth.msu.edu \end{center} \begin{abstract} We survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is is based on Zaslavsky's theory of signed graphs. The second approach is algebraic and employs results of Saito and Terao about free hyperplane arrangements. Finally we consider a purely combinatorial theorem of Stanley about supersolvable lattices and its generalizations. \end{abstract} \pagestyle{plain} \section{Introduction} \label{i} If one chooses a random polynomial with real coefficients, the chances are very small that it will have all its roots in $\bbZ_{\ge0}$, the nonnegative integers. However if one considers the characteristic polynomials of various lattices that arise in practice, a surprisingly large number of them do factor over $\bbZ_{\ge0}$. The natural question to ask is: Why? In this survey, we provide three reasons with tools drawn from three different areas of mathematics: graph theory/geometry, algebra, and pure combinatorics. The first of these uses Zaslavsky's lovely theory of signed graph coloring~\cite{zas:grs,zas:sgc,zas:cis} which can be generalized to counting points of $\bbZ^n$ inside a certain polytope~\cite{bs:cep}. The next technique is based on theorems of Saito~\cite{sai:tld} and Terao~\cite{ter:gef} about free hyperplane arrangements. Work has also been done on related concepts such as inductive freeness~\cite{ter:ahf} and recursive freeness~\cite{zie:ach}. The third method employs a theorem of Stanley~\cite{sta:sl} on semimodular supersolvable lattices which has recently been generalized by Blass and myself~\cite{bs:mfl} by relaxing both restrictions on the lattice. Since this paper is expository, I will provide a fair number of definitions and examples. However for the proofs of most theorems the reader will have to see the articles cited. A {\it lattice}, $L$, is a set with a partial order $\le$ such that every pair $x,y\in L$ has a meet or greatest lower bound, $x\mt y$, and a join or least upper bound, $x\jn y$. All our lattices will be finite and so have a unique minimal element $\zh=\Mt L$ and a unique maximal element $\oh=\Jn L$. Another set of important elements of $L$ are its {\it atoms\/} which are all elements $a$ covering $\zh$. (If $x,y\in L$ then {\it $x$ covers $y$\/} if $x>y$ and there is no $z$ with $x>z>y$.) We let $A(L)$ denote the atom set of $L$. One of the fundamental invariants of a lattice, or indeed of any partially ordered set, is its {\it M\"obius function of $L$}, $\mu: L\ra \bZ$, defined recursively by \beq \label{mu} \mu(x)=\case{1}{if $x=\zh$,}{-\sum_{y\zh$.} \eeq The number-theoretic M\"obius function is obtained as a special case by taking $L$ to be the lattice of divisors of an integer ordered by divisibility. Our main object of study will be the generating function for $\mu$. Let $L$ be {\it ranked\/} so that for any $x\in L$ all maximal chains from $\zh$ to $x$ have the same length denoted $\rho(x)$ and called the {\it rank of $x$}. (A {\it chain\/} is a totally ordered subset of $L$.) The {\it characteristic polynomial of $L$\/} is then \beq \label{chi1} \chi(L,t)=\sum_{x\in L} \mu(x) t^{\rho(\oh)-\rho(x)}. \eeq Note that we use the corank rather than the rank in the exponent on $t$ so that $\chi$ will be monic. The usual generating function by rank is the related {\it Poincar\'e polynomial of $L$} $$\pi(L,t))=\sum_{x\in L} |\mu(x)| t^{\rho(x)}.$$ \thicklines \setlength{\unitlength}{2pt} \bfi \btab{lll} \bpi(20,90)(-10,-20) \GaaL{-20pt}{0} \GacL{-20pt}{1} \GaeL{-20pt}{2} \GagL{-20pt}{3} \Gaaac \Gacae \Gaeag \put(0,-20){\makebox(0,0){The chain $C_3$}} \epi& \bpi(90,90)(-10,-20) \Gda \put(30,-10){\makebox(0,0){$\emp$}} \put(30,-20){\makebox(0,0){The Boolean algebra $B_3$}} \GacL{-25pt}{$\{1\}$} \GdcL{-25pt}{$\{2\}$} \GgcL{-25pt}{$\{3\}$} \GaeL{-35pt}{$\{1,2\}$} \GdeL{-35pt}{$\{1,3\}$} \GgeL{-35pt}{$\{2,3\}$} \Gdg \put(30,70){\makebox(0,0){$\{1,2,3\}$}} \Gdaac \Gdadc \Gdagc \Gacae \Gacde \Gdcae \Gdcge \Ggcde \Ggcge \Gaedg \Gdedg \Ggedg \epi& \bpi(70,90)(-10,-20) \Gdb \put(30,0){\makebox(0,0){$\{1\}\{2\}\{3\}$}} \put(30,-20){\makebox(0,0){The Partition lattice $\Pi_3$}} \GadL{-50pt}{$\{1,2\}\{3\}$} \GddL{-50pt}{$\{1,3\}\{2\}$} \GgdL{-50pt}{$\{1\}\{2,3\}$} \Gdf \put(30,60){\makebox(0,0){$\{1,2,3\}$}} \Gdbad \Gdbdd \Gdbgd \Gaddf \Gdddf \Ggddf \epi \etab \capt{Some example lattices} \label{ex} \efi Let us look at three simple examples of lattices and their characteristic polynomials. The {\it chain $C_n$\/} consists of the integers $\{0,1,\ldots,n\}$ ordered in the usual manner, see Figure~\ref{ex} for a picture of $C_3$. It is immediate directly from the definition~(\ref{mu}) that in $C_n$ we have $$\mu(x)={\left\{\barr{ll}1&\mbox{if $x=0$}\\ -1&\mbox{if $x=1$}\\ 0&\mbox{if $x\ge2$} \earr\right.}$$ and so $$\chi(C_n,t)=t^n-t^{n-1}=t^{n-1}(t-1).$$ The {\it Boolean algebra $B_n$\/} has as elements all subsets of $[n]:=\{1,2,\ldots,n\}$ and $\sbe$ as order relation, the case $n=3$ being displayed in Figure~\ref{ex}. It is well known that for $x\in B_n$ we have $\mu(x)=(-1)^{|x|}$ were the absolute value signs denote cardinality. It follows that $$\chi(B_n,t)=\sum_{x\sbe[n]}(-1)^{|x|}t^{n-|x|}=(t-1)^n.$$ Finally consider the {\it partition lattice $\Pi_n$\/} which consists of all partitions of $[n]$ ordered by refinement. Direct computation with $\Pi_3$ as shown in Figure~\ref{ex} shows that $\chi(\Pi_3,t)=t^2-3t+2=(t-1)(t-2)$. In general $$\chi(\Pi_n,t)=(t-1)(t-2)\cdots(t-n+1).$$ Note that in all three cases $\chi$ has only nonnegative integral roots. Many of our example lattices will arise as intersection lattices of subspace arrangements. (See~\cite{bjo:sa,ot:ah} for details about the theory of arrangements.) A {\it subspace arrangement\/} is a finite set \beq \label{cA} \cA=\{K_1,K_2,\ldots,K_l\} \eeq of subspaces of real Euclidean space $\bbR^n$. If $\dim K_i=n-1$ for $1\le i\le l$ then we say that $\cA$ is a {\it hyperplane arrangement} and will uses $H$'s in place of $K$'s. The {\it intersection lattice of $\cA$}, $L(\cA)$, has as elements all subspaces $X$ of $\bbR^n$ that can be written as an intersection of some of the elements of $\cA$. The partial order is reverse inclusion, so that $X\le Y$ if and only if $X\spe Y$. So $L(\cA)$ has minimal element $\bbR^n$, maximal element $K_1\cap\cdots\cap K_l$, and join operation $X\jn Y=X\cap Y$. The {\it characteristic polynomial of $A$\/} is defined by \beq \label{chi2} \chi(\cA,t)=\sum_{X\in L(\cA)} \mu(X) t^{\dim X}. \eeq This is not necessarily the same as $\chi(L(\cA),t)$ as defined in~(\ref{chi1}). If $\cA$ is a hyperplane arrangement then the two will be equal up to a factor of a power of $t$, so from the point of view of having integral roots there is no difference. In the general subspace case~\ree{chi1} and~\ree{chi2} may be quite dissimilar and often the latter turns out to factor at least partially over $\bbZ_{\geq0}$ while the former does not. In the arrangement case the roots of~\ree{chi2} are called the {\it exponents of $\cA$} and denoted $\exp \cA$. In fact when $\cA$ is a the set of reflecting hyperplanes for a Weyl group $W$ then these roots are just the usual exponents of the $W$~\cite{ter:gef} which are always nonnegative integers. All three of our previous example lattices can be realized as intersection lattices of subspace arrangements. The $n$-chain is $L(\cA)$ with $\cA=\{K_0,\ldots,K_n\}$ where $K_i$ is the set of all points having the first $i$ coordinates zero. The Boolean algebra is the intersection lattice of the arrangement of coordinate hyperplanes $H_i:\ x_i=0$, $1\le i\le n$. To get the partition lattice we use the {\it Weyl arrangement of type $A$} $$ \cA_n=\{x_i-x_j=0\ :\ 1\le i