Compound Pathway Problem with holes

Question

The picture below is what I am trying to create.

I have asked a similar question before, and I got an excellent answer, but it's still not 100% what I am looking for. I know that I have to use \usepackage{tikz} and possibly \usepackage{xintexpr} and \usetikzlibrary{math}

\documentclass[tikz,border=5]{article} \usepackage{tikz} \usepackage{xintexpr} \usetikzlibrary{math} \begin{document}

\begin{tikzpicture}[x=2cm,y=-2cm, node 0/.style={fill=red!20}]
\tikzmath{%
  int \i, \j, \m, \n, \t;
  \m = 5; \n = 5;
  % Initialise board.
  for \i in {0,...,\m}{
  for \j in {0,...,\n}{
    \t{\i,\j} = 0;
         };
       };
  % Create holes.
  \t{2,3} = -1;
  \t{3,4} = -1;
  \t{5,2} = -1;
 % Perform calculations.
  for \i1 in {0,...,\m}{
    for \j1 in {0,...,\n}{
      if (\t{\i1,\j1} == -1) then {
        \t{\i1,\j1} = 0;
      } else {
         if (\i1 == 0 || \j1 == 0) then  {
          \t{\i1,\j1} = 1;            
      } else {
         \i2 = \i1 - 1;
         \j2 = \j1 - 1;
         \t{\i1,\j1} = \t{\i2,\j1} + \t{\i1,\j2}; 
            };
          };
        };
      };
 % Draw nodes.
    for \i1 in {0,...,\m}{
     for \j1 in {0,...,\n}{
     { \node [circle, fill=blue!20, minimum size=1cm, node \t{\i1,\j1}/.try] 
         (n-\i1-\j1) at (\j1, \i1) {\t{\i1,\j1}}; };
        };   
       };
  % Draw edges.
for \i1 in {0,...,\m}{
  for \j1 in {0,...,\n}{
    \i2 = \i1 + 1;
    \j2 = \j1 + 1;
       if (\i1 < \m) then {
         if (\t{\i2,\j1} > 0) then { 
               { \draw [thick, -stealth] (n-\i1-\j1) -- (n-\i2-\j1); };
      }; 
    };
    if (\j1 < \n) then {
    if (\t{\i1,\j2} > 0) then { 
          { \draw [thick, -stealth] (n-\i1-\j1) -- (n-\i1-\j2); };
        }; 
       };
     };
   };
  }
  \end{tikzpicture}

\end{document}

This code has the idea to what I am looking for, but what I want is to refine the code so it can look exactly like the picture included. I'm new to latex and the idea of programming/coding, but I am trying to learn.

Also my picture needs to fit into my homework problem setting

\documentclass[28pt]{article}
\usepackage{fancyhdr}
\usepackage[includeheadfoot,margin=1.0cm]{geometry}

Answer 1

Ok, so an additional data structure is required to model the "routes" which have holes.

\documentclass[border=5]{standalone}
\usepackage{tikz}
\usetikzlibrary{math}
\begin{document}
\begin{tikzpicture}[x=2cm, y=-2cm]
\tikzmath{%
  int \i, \j, \m, \n, \t, \r;
  \m = 6; \n = 6;
  % Initialise board.
  for \i in {0,...,\m-1}{
    for \j in {0,...,\n-1}{
      \t{\i,\j} = 0;
    }; 
  };
  for \i in {0,...,\m-1}{
    for \j in {0,...,\n-1}{
      \r{\i,\j} = 11;
    }; 
  };
  % Starting point.
  \t{0,0} = 1;
  % Create holes in route.
  % 01 can go east
  % 10 can go south
  % 11 can go east and south
  \r{2,2} = 10;
  \r{3,3} = 10;
  \r{4,2} = 01;
  % Perform calculations.
  for \i1 in {0,...,\m-1}{
    for \j1 in {0,...,\n-1}{
      if (\i1 == 0 && \j1 == 0) then {
        \t{0,0} = 1;
      } else {
        if (\j1 > 0) then {
            \j2 = \j1 - 1;
            if (mod(\r{\i1,\j2}, 2) == 1) then {
                \t{\i1,\j1} = \t{\i1,\j1} + \t{\i1,\j2};
            };
        };
        if (\i1 > 0) then {
            \i2 = \i1 - 1;
            if (mod(floor(\r{\i2,\j1} / 10), 2) == 1) then {
                \t{\i1,\j1} = \t{\i1,\j1} + \t{\i2,\j1};
            };
        };
      };
    };
  };
 % Draw nodes.
    for \i1 in {0,...,\m-1}{
     for \j1 in {0,...,\n-1}{
     { \fill  (\j1, \i1) circle [radius=0.05]
        node [above left] {\t{\i1,\j1}}; };
      };   
       };
    % Draw edges.
    for \i1 in {0,...,\m-1}{
      for \j1 in {0,...,\n-1}{
        if (mod(\r{\i1,\j1}, 2) == 1 && \j1 < \n - 1) then {
           { \draw [thick, -stealth, shorten >=0.1cm] (\j1,\i1) -- ++(1, 0); };
        }; 
        if (mod(floor(\r{\i1,\j1} / 10), 2) == 1 && \i1 < \m - 1) then {
           { \draw [thick, -stealth,  shorten >=0.1cm] (\j1,\i1) -- ++(0,1); };
        }; 
      };
   };
}
\end{tikzpicture}
\end{document}

Answer 2

I hope this is what you want

\documentclass{standalone}
\usepackage{tikz}
\usepgfmodule{parser}
\begin{document}

\catcode`\^^`=12\relax\pgfparserdef{countpath}{initial}{the character ^^`}{\nextcell}\catcode`\^^`=10\relax
\catcode`\^^M=12\relax\pgfparserdef{countpath}{initial}{the character ^^M}{\nextline}\catcode`\^^M=5\relax
                      \pgfparserdef{countpath}{all}{the character ;}{\endcountpath}
                      \pgfparserdef{countpath}{all}{the character +}{\nextcell\drawanode}
                      \pgfparserdef{countpath}{all}{the character |}{\nextcell\draw(\coordx,-\coordy)+(0,1)--+(0,-1);\expandafter\xdef\csname cp/\coordx/\coordy\endcsname{C}}
                      \pgfparserdef{countpath}{all}{the character -}{\nextcell\draw(\coordx,-\coordy)+(-1,0)--+(1,0);\expandafter\xdef\csname cp/\coordx/\coordy\endcsname{C}}
\def\countpath{
    \begin{tikzpicture}[shorten <=9pt,shorten >=9pt]
        \def\coordx{0}\def\coordy{0}
        \catcode`\^^M=12\relax\catcode`\^^`=12\relax
        \pgfparserparse{countpath}%
}
\def\endcountpath{
    \end{tikzpicture}
    \pgfparserswitch{final}
}
\def\nextcell{\pgfmathtruncatemacro\coordx{\coordx+1}}
\def\nextline{\pgfmathtruncatemacro\coordy{\coordy+1}\def\coordx{0}}
\def\drawanode{
    \def\numberofpaths{0}
    \ifnum\coordy=1
        \ifnum\coordx=1
            \def\numberofpaths{1}
        \fi
    \fi
    % check upper neighbor
    \pgfmathtruncatemacro\coordyminusi{\coordy-1}
    \expandafter\if\csname cp/\coordx/\coordyminusi\endcsname\relax\else
        \pgfmathtruncatemacro\coordyminusii{\coordy-2}
        \pgfmathtruncatemacro\numberofpaths{\numberofpaths+\csname cp/\coordx/\coordyminusii\endcsname}
    \fi
    % check left neighbor
    \pgfmathtruncatemacro\coordxminusi{\coordx-1}
    \expandafter\if\csname cp/\coordxminusi/\coordy\endcsname\relax\else
        \pgfmathtruncatemacro\coordxminusii{\coordx-2}
        \pgfmathtruncatemacro\numberofpaths{\numberofpaths+\csname cp/\coordxminusii/\coordy\endcsname}
    \fi
    \expandafter\xdef\csname cp/\coordx/\coordy\endcsname{\numberofpaths}
    \draw(\coordx,-\coordy)node{\numberofpaths};
}

\countpath
+-+-+-+-+-+-+-+ +-+
| |   | | |   | | |
+-+ +-+-+ +-+-+-+-+
|   | | |   | | | |
+-+-+-+-+-+-+-+ +-+
| | | | |     | | |
+-+ +-+-+ +-+-+ +-+
|   | | |   | | | |
+-+-+-+-+-+-+-+-+-+
| |   | | |   | | |
+-+-+-+-+-+-+-+ +-+
|   | | | | | | | |
+-+-+-+-+-+-+-+-+-+
;

\end{document}