答案1
一个主意：
\documentclass[border=2mm]{standalone}
\usepackage{tikz}
\newcommand\catalannumber[3]{
% start point, size, Dyck word (size x 2 booleans)
\fill[cyan!25] (#1) rectangle +(#2,#2);
\fill[fill=lime]
(#1)
\foreach \dir in {#3}{
\ifnum\dir=0
-- ++(1,0)
\else
-- ++(0,1)
\fi
} |- (#1);
\draw[help lines] (#1) grid +(#2,#2);
\draw[dashed] (#1) -- +(#2,#2);
\coordinate (prev) at (#1);
\foreach \dir in {#3}{
\ifnum\dir=0
\coordinate (dep) at (1,0);
\else
\coordinate (dep) at (0,1);
\fi
\draw[line width=2pt,-stealth] (prev) -- ++(dep) coordinate (prev);
};
}
\begin{document}
\begin{tikzpicture}
\catalannumber{0,0}{4}{0,1,0,0,1,1,0,1};
\catalannumber{0,-9}{8}{0,1,0,0,1,1,0,0,0,0,1,1,0,1,1,1};
\end{tikzpicture}
\end{document}
含所有 Dyck 编号的版本：
\documentclass{article}
\usepackage[margin=1cm]{geometry}
\usepackage{tikz}
\usepackage{ifthen}
\pagestyle{empty}
\newcommand\catalannumber[3]{
% start point, size, Dyck word (size x 2 booleans)
\fill[cyan!25] (#1) rectangle +(#2,#2);
\fill[fill=lime]
(#1)
\foreach \dir in {#3}{
\ifnum\dir=0
-- ++(1,0)
\else
-- ++(0,1)
\fi
} |- (#1);
\draw[help lines] (#1) grid +(#2,#2);
\draw[dashed] (#1) -- +(#2,#2);
\coordinate (prev) at (#1);
\foreach \dir in {#3}{
\ifnum\dir=0
\coordinate (dep) at (1,0);
\tikzset{label/.style={below}};
\else
\coordinate (dep) at (0,1);
\tikzset{label/.style={left}};
\fi
\draw[line width=2pt,-stealth] (prev) --
++(dep) coordinate (prev);
};
}
\newcommand\genallcatalannumbers[6]{%
\pgfmathtruncatemacro{\steps}{#4+#5}%
\ifthenelse{#3>\steps}{%
\ifthenelse{#5<#4}{%
{%
\ifthenelse{#4<#2}{%
\pgfmathtruncatemacro{\nbzero}{#4+1}%
\genallcatalannumbers{#1}{#2}{#3}{\nbzero}{#5}{#6,0}%
}{}%
}%
{%
\pgfmathtruncatemacro{\nbone}{#5+1}%
\genallcatalannumbers{#1}{#2}{#3}{#4}{\nbone}{#6,1}%
}%
}{%
\pgfmathtruncatemacro{\nbzero}{#4+1}%
\genallcatalannumbers{#1}{#2}{#3}{\nbzero}{#5}{#6,0}%
}%
}{%
\begin{tikzpicture}[myscale]
\catalannumber{#1}{#2}{#6}
\end{tikzpicture} %
}%
}
\newcommand\allcatalannumbers[2]{%
\pgfmathtruncatemacro{\nbsteps}{2*#2}%
\genallcatalannumbers{#1}{#2}{\nbsteps}{1}{0}{0}%
}
\begin{document}
\begin{enumerate}
\foreach \num in {1,...,5}{
\item%{\LARGE\bfseries \num}
\tikzset{myscale/.style={scale=2/\num}}
\noindent\allcatalannumbers{0,0}{\num}\par
}
\end{enumerate}
\end{document}
答案2
砷钛矿钾Z 的运行速度比 PSTricks 慢，而且我不知道如何在 PSTricks 中编写递归函数，我被迫使用通过\write18宏从 LaTeX 内部执行的外部应用程序。
如果你编译以下 C# 源代码
// This is CatalanLocator.cs
using System.Collections.Generic;
using System.IO;
namespace CatalanLocator
{
class Program
{
static void Populate(List<bool> set, int m, int n)
{
if (m == 0)
{
if (n == 0)
{
using (StreamWriter sw = new StreamWriter("data.txt", true))
{
sw.Write("{");
int x;
for (x = 0; x < set.Count - 1; x++)
sw.Write("{0},", set[x] ? 1 : 0);
sw.WriteLine("{0}}}", set[x] ? 1 : 0);
}
}
else
{
List<bool> temp = new List<bool>(set);
temp.Add(true);
Populate(temp, m, n - 1);
}
}
else
{
if (m == n)
{
List<bool> temp = new List<bool>(set);
temp.Add(false);
Populate(temp, m - 1, n);
}
else
{
List<bool> temp1 = new List<bool>(set);
temp1.Add(false);
Populate(temp1, m - 1, n);
List<bool> temp2 = new List<bool>(set);
temp2.Add(true);
Populate(temp2, m, n - 1);
}
}
}
static void Main(string[] args)
{
int N = int.Parse(args[0]);
Populate(new List<bool>(), N, N);
}
}
}
，csc CatalanLocator.cs您将获得一个名为的可执行文件CatalanLocator.exe。
CatalanLocator 6例如，从 DOS 提示符手动执行将生成一个名为 的文本文件data.txt。它包含多行二进制数集。每组代表一条 Catalan 路径。（Catalan 路径是我自己发明的术语。维基百科可能没有这样的定义。）
不过，我更喜欢从 LaTeX 输入文件执行，CatalanLocator如下面的代码所述：
% CatalanDiagram.tex
\documentclass{article}
\usepackage{pstricks-add}
\usepackage[active,tightpage]{preview}
\PreviewBorder=12pt
\PreviewEnvironment{pspicture}
\newcounter{oX}% old x
\newcounter{oY}% old y
\newcounter{nX}% new x
\newcounter{nY}% new y
\newcounter{N}% N by N grid
\newcommand\CatalanPath[2][-]
{
% reset all counters
\setcounter{nX}{0}\setcounter{nY}{0}\setcounter{oX}{0}\setcounter{oY}{0}
% please make sure there is no blank line allowed in \psforeach
\psforeach{\i}{#2}
{
\ifnum\i=0
\stepcounter{nX}% move to the right
\else
\stepcounter{nY}% move upward
\fi
%
% draw a single segment
\psline[arrows=#1,linewidth=1pt](\theoX,\theoY)(\thenX,\thenY)
%
% renew the old counters
\setcounter{oX}{\value{nX}}
\setcounter{oY}{\value{nY}}
}
}
\newcommand\CatalanDiagram[1]
{
% probe the given Catalan's set and find the corresponding grid dimension
\setcounter{N}{0}\psforeach{\i}{#1}{\stepcounter{N}}\setcounter{N}{\numexpr\theN/2\relax}
\begin{pspicture}(\theN,\theN)
{
\psset{fillstyle=solid,linestyle=none,opacity=0.5}
% fill the upper region
\pscustom[fillcolor=cyan]{\CatalanPath{#1}\psline(\thenX,\thenY)(0,\thenY)}
% fill the lower region
\pscustom[fillcolor=yellow]{\CatalanPath{#1}\psline(\thenX,\thenY)(\thenX,0)}
}
% draw grid
\psgrid[style=gridstyle,gridlabels=0pt]
% draw a diagonal line
\psline[linestyle=dashed,dash=3pt 1.5pt,linecolor=red](\theN,\theN)
% draw Catalan path
\CatalanPath[->]{#1}
\end{pspicture}
}
\newread\myfile
\def\temp#1{\CatalanDiagram{#1}}
\begin{document}
\immediate\write18{cmd /C del data.txt}
\immediate\write18{CatalanLocator.exe 6}
\openin\myfile=data.txt\relax
\loop
\read\myfile to \j
\unless\ifeof\myfile
\expandafter\temp\j\relax
\repeat
\closein\myfile
\end{document}
执行latex --shell-escape CatalanDiagram后接着dvips CatalanDiagram执行会ps2pdf CatalanDiagram.ps产生CatalanDiagram.pdf。
以下 GIF 图像是动画6×6加泰罗尼亚图。我故意把它弄得很小以节省您的带宽。
答案3
只是为了好玩...基于 PolGab 的第一个答案的随机版本。
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\pgfdeclarelayer{mylayer}
\pgfsetlayers{background,mylayer,main}
\usepackage{etoolbox}
% counters to check movements
\newcounter{up}
\newcounter{rightp}
\newcommand\catalannumber[2]{
\setcounter{up}{0}
\setcounter{rightp}{0}
% start point, size
\begin{pgfonlayer}{background}
\fill[cyan!25] (#1) rectangle +(#2,#2);
\end{pgfonlayer}
\draw[help lines] (#1) grid +(#2,#2);
\draw[dashed] (#1) -- +(#2,#2);
\coordinate (prev) at (#1);
\pgfmathtruncatemacro\dim{#2*2}
\foreach \x in {1,...,\dim}{
\pgfmathtruncatemacro\dir{round(rand)}
% first case
\ifnum\x=1
\pgfmathtruncatemacro\dir{0}
\fi
% normal behaviour
\ifnumodd{\x}{
% check number of up to not exceed the diagonal
\pgfmathtruncatemacro\numupadmitted{\x/2}%
\ifnum\numupadmitted=\theup
\pgfmathtruncatemacro\dir{0}
\fi
}{}
% check number of rightp to not exceed the border
\ifnum\therightp=#2
\pgfmathtruncatemacro\dir{1}
\fi
% movements
\ifnum\dir=0
\coordinate (dep) at (1,0);
\stepcounter{rightp}
\else
\coordinate (dep) at (0,1);
\stepcounter{up}
\fi
\draw[line width=2pt,-stealth] (prev) -- ++(dep) node(a\x){} coordinate (prev){};
}
\begin{pgfonlayer}{mylayer}
\fill[orange!25](#1)
\foreach \x in {1,...,\dim}{
--(a\x.center)
}|-(#1);
\end{pgfonlayer}
}
\begin{document}
\begin{tikzpicture}
\catalannumber{0,0}{7};
\catalannumber{0,-10}{9};
\end{tikzpicture}
\end{document}
答案4
纯粹是为了我自己的娱乐，我制作了一个metapost版本，绘制的图表与 wiki 页面上的其他图像之一相映成趣。
虽然在我的计算机上生成所有长度为 20 的 Dyck 单词需要大约一分钟的时间，但最终它确实做到了，并且
\begin{mplibcode}
beginfig(0);
dwords(10);
draw dyckdiagram(10,4500);
endfig;
\end{mplibcode}
生产
代码（编译用lualatex）：
\documentclass{article}
\usepackage{luamplib}
\mplibnumbersystem{double}
\mplibtextextlabel{enable}
\mplibcodeinherit{enable}
\mplibforcehmode
\begin{document}
\begin{center}
\begin{mplibcode}
color rred,bblue;
rred:=(1.0,.377,.398);
bblue:=(0.37,0.57,0.7);
% http://www.acta.sapientia.ro/acta-info/C1-1/info1-9.pdf
vardef dwords(expr n)=% generates positions of down steps
for i=1 upto n: b[i]:=2i; endfor;
cnt:=0;
forever:
cnt:=cnt+1;
for i=1 upto n: D[cnt][b[i]]:=1; endfor; % save down steps
ind:=0; % haven't bothered to figure out how the following works
for i=n-1 downto 1:
if b[i]<n+i:
b[i]:=b[i]+1;
for j=i+1 upto n-1:
if b[j-1]+1<2j:
b[j]:=2j;
else:
b[j]:=b[j-1]+1;
fi;
endfor;
ind:=1;
exitif true;
fi;
endfor;
exitif ind=0;
endfor;
enddef;
% stores path
vardef dpath(expr n,m)=
save hstep_,vstep_,h_; vstep_=hstep_:=0; h:=sqrt(3)/2;
(0,0)
for i= 1 upto 2n:
if not known D[m][i]:
--((incr hstep_)/2,(incr vstep_)*h)
else:
--((incr hstep_)/2,(decr vstep_)*h)
fi
endfor
enddef;
% n=half-order, m=word number
vardef dyckdiagram(expr n,m)=
save p_,q_,g_,d_,h_,pic_; path p_,q_; picture pic_; h_:=sqrt(3)/2;
pic_:=image(
for i=n downto 1: % triangular grid
for j=1 upto i:
fill fullcircle scaled .5u shifted (u*(.5j+(n-i),h_*j)) withcolor .7;
endfor;
endfor;
d_:=0;% number of down steps
for i=2 upto 2n:
if known D[m][i]: % if down
d_:=d_+1;
if not known D[m][i-1]: % and previous was up
fill fullcircle scaled .5u shifted (u*(.5(i-1), (i-2d_+1)*h_)) withcolor rred;
fi;
fi;
endfor;
draw dpath(n,m) scaled u withpen pencircle scaled .2u withcolor rred;
r:=sqrt(3)/2;
% for rounded outlines
path p[]; p[-1]:= (1,0) for i=1 upto 120: --(1,0) rotated 3i endfor--cycle;
p[0]:=(.5,r)--(.5*(2n-1),r)--(n/2,n*r)--cycle;
pen bpen; bpen:=makepen(p[-1]); pickup bpen;
p[1]:=envelope bpen of p[0];
draw p[1] scaled u withpen pencircle scaled .2u withcolor bblue;
);
pic_
enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dwords(4);
beginfig(0);
u:=.3cm;
for i=1 upto 3: draw dyckdiagram(4,i) shifted ((i-1)*6u,0); endfor;
endfig;
\end{mplibcode}\\[5pt]
\begin{mplibcode}
beginfig(0);
for i=4 upto 7: draw dyckdiagram(4,i) shifted ((i-4)*6u,0); endfor;
endfig;
\end{mplibcode}\\[5pt]
\begin{mplibcode}
beginfig(0);
for i=8 upto 11: draw dyckdiagram(4,i) shifted ((i-8)*6u,0); endfor;
endfig;
\end{mplibcode}\\[5pt]
\begin{mplibcode}
beginfig(0);
for i=12 upto 14: draw dyckdiagram(4,i) shifted ((i-12)*6u,0); endfor;
endfig;
\end{mplibcode}
\end{center}
%\begin{mplibcode}
%beginfig(0);
% dwords(10);
% draw dyckdiagram(10,4500);
%endfig;
%\end{mplibcode}
\end{document}