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Cell[CellGroupData[{
Cell["45-degree square gas experiment", "Subsection"],

Cell["\<\
Consider a single 45-degree rotated square in an N by M octagon.  I want to \
count for each lattice site the number of ways that the rotated square might \
end up in this cell.  I will paths that would end at this site for the \
45-degree rhombus strings and multiply.\
\>", "Text"],

Cell[BoxData[{
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Cell["\<\
The Gessel-Viennot algorithm is actually unnecessary here. I use simple \
combinations.\
\>", "Text"],

Cell[BoxData[
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                  1)\), \ \((i\  - \ 1)\)]\ Binomial[\((\ 
                Width\  - \ i)\)\  + \((\ j - 1)\), \((\ 
              j - 1)\)]\ \ Binomial[\((\ 
                Width\  - \ i)\)\  + \((Height\  - \ j)\), \((Height\  - \ 
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            }], "\[NoBreak]", ")"}],
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Cell["\<\

This represents the number of distinct choices of the four 45-degree rhombus \
strings that can leave the single 45-degree rotated square at this site, \
indicating the degree of repulsion from nearby hard walls.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Implementation of Destinville's formula for Octagon tilings", \
"Subsection"],

Cell["\<\

This is a very, very brute force algorithm.  The key to doing it properly \
lies in the enumeration of the squares.\
\>", "Text"],

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          GetConventionX[x, i, u - 1, a, b, c, d]\ \  + \ 
          GetConventionY[y, j, u, a, b, c, d]\  - \ 
          GetConventionY[y, i, u - 1, a, b, c, d], \ 
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          GetConventionX[x, i, u - 1, a, b, c, d]\ \  + \ j\  - \ i]; 
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                            a, b, c, d, \ 1, \ 
                            1}]\)[\([1]\)]\)[\([1]\)]\)[\([1]\)]\)[\([1]\)]\)[\
\([1]\)]\)[\([1]\)]\)[\([1]\)]]\ ;\)\), "\[IndentingNewLine]", 
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                            a, b, c, d, \ 1, \ 
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          d_] := \((\[Product]\+\(u = 1\)\%\(d + 1\)DeterminantM[u, \ x, \ y, 
                a, b, c, 
                d]\ )\) \((\[Product]\+\(v = 1\)\%\(b + 1\)DeterminantP[v, \ 
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Cell[TextData[{
  "\nThis should define the necessary prerequisites to get the path product:  \
now I must sum over all chocies of x and y arrays. The items in x run from 0 \
to a, and the items in y run from 0 to c.  They also must satisfy the \
conditions from Destainville's paper.  This is then an inherently brutal \
algorithm that I will deal with in a brutal way.  \n\nFirst, I need to know \
if my selected X and Y are actually in the appropriate cross-product space.  \
\n\nThere is a faster algorithm to check this:  I could hypothetically check \
in N time, but it's harder to program in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " than C++ since in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " I must nest everything in parameters."
}], "Text"],

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Cell["\<\
Now I need to actually construct all possible matrices (b by d), x having \
values 0 through a, y having values 0 through c.\
\>", "Text"],

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            Return[\ ForAllMatricesY[1, \ 1, \ inMatrix, \ 
                Array[Zero, \ {b, \ d}], \ a, \ b, \ c, \ d]], \ \ Return[
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            Return[ForAllMatricesX[px\  + \ 1, \ 1, \ inMatrix\ , \ \ a, \ 
                b, \ c, \ d]], \[Sum]\+\(n = 0\)\%a ForAllMatricesX[px, \ 
                py\  + \ 1, \ 
                inMatrix + \ 
                  SparseArray[{{px, \ py}}\  \[Rule] \ {n}, \ {b, 
                      d}], \ \ a, \ b, \ c, \ 
                d]\ ]];\)\), "\[IndentingNewLine]", 
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          d_]\  := \ If[And[px \[Equal] b, \ py \[Equal] d + 1], 
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            Return[\ PathProduct[inMatrixX, \ inMatrix, a, b, c, 
                d]], \ \ Return[0]], \ 
          If[And[px\  < \ b, \ py\  \[Equal] \ d\  + \ 1]\ , \ 
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                inMatrix\ , \ \ a, \ b, \ c, \ d]], \[Sum]\+\(n = 0\)\%c 
                ForAllMatricesY[px, \ py\  + \ 1, 
                inMatrixX, \ \ inMatrix + \ 
                  SparseArray[{{px, \ py}}\  \[Rule] \ {n}, \ {b, 
                      d}], \ \ a, \ b, \ c, \ d]\ ]];\)\)}], "Input"],

Cell[BoxData[
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