hangingStrings[n_,a_List]:=2n-(PadLeft[a,Length[a]+1]+PadRight[a,Length[a]+1])

hangingStrings[8,{1,2,3}]

annularConsequences[n_,a_List,m_]:=Binomial[2m,m-(Plus@@hangingStrings[n,a])/2]

Binomial

Subsets[Range[6],{2}]

cr[m_,x_]:=Module[{a},
a[0]=0;
(a[#]=a[#-1]+1)&/@x;
a[n_]:=a[n]=a[n-1]+If[a[n-1]>0,-1,0];
a[m]
]

rectangularConsequences[m_Integer,k_Integer]/;0≤k≤m:=rectangularConsequences[m,k]=Length[Cases[Subsets[Range[2m],{m-k}],s_/;cr[2m,s]==0]]

Table[rectangularConsequences[m,k],{m,1,8},{k,0,m}]//TableForm

1,5,20,75,275,1001

1,9,35,75,90,42

Clear[rc]

rc[m_,k_]/;m>0∧k>0:=rc[m,k]=rc[m-1,k-1]+rc[m-1,k+1]
rc[m_,0]:=rc[m,0]=rc[m-1,1]
rc[0,0]=1;
rc[0,_]:=0

rc[0,2]

Clear[rectangularConsequences]

rectangularConsequences[m_,k_]:=rc[2m,2k]

rectangularConsequences[m_Integer,k_Integer]/;k<0∨k>m:=0

rectangularConsequences[n_Integer,a_List,m_Integer]:=rectangularConsequences[m,(Plus@@hangingStrings[n,a])/2]

rectangularConsequences[8,{},9]

rectangularConsequences[8,{7},9]

annularConsequences[8,{},10]

rectangularConsequences[8,{},10]

rectangularConsequences[8,{7},10]

rectangularConsequences[8,{6},10]

rectangularConsequences[8,{7,7},10]

annularConsequences[8,{},11]

rectangularConsequences[8,{},11]

rectangularConsequences[8,{7},11]

rectangularConsequences[8,{6},11]

rectangularConsequences[8,{5},11]

rectangularConsequences[8,{6,7},11]

rectangularConsequences[8,{7,6},11]

1309+209+21+1

annularConsequences[8,{},12]

rectangularConsequences[8,{},12]

rectangularConsequences[8,{7},12]

rectangularConsequences[8,{6},12]

rectangularConsequences[8,{5},12]

rectangularConsequences[8,{4},12]

rectangularConsequences[8,{6,6},12]

rectangularConsequences[8,{5,6,5},12]

8602+1748+252+23+1

annularConsequences[8,{},13]

rectangularConsequences[8,{},13]

rectangularConsequences[8,{7},13]

rectangularConsequences[8,{6},13]

rectangularConsequences[8,{5},13]

rectangularConsequences[8,{4},13]

rectangularConsequences[8,{3},13]

rectangularConsequences[8,{5,6},13]

rectangularConsequences[8,{4,7},13]

50830+12350+2275+299+25+1

annularConsequences[8,{},14]

rectangularConsequences[8,{},14]

rectangularConsequences[8,{7},14]

rectangularConsequences[8,{6},14]

rectangularConsequences[8,{5},14]

rectangularConsequences[8,{4},14]

rectangularConsequences[8,{3},14]

rectangularConsequences[8,{2},14]

rectangularConsequences[8,{5,5},14]

rectangularConsequences[8,{4,6},14]

rectangularConsequences[8,{3,7},14]

278460+77805+17199+2898+350+27+1

annularConsequences[8,{},15]

rectangularConsequences[8,{},15]+Plus@@(rectangularConsequences[8,{#},15]&/@Range[7])

rectangularConsequences[8,{4,5},15]

rectangularConsequences[8,{3,6},15]

rectangularConsequences[8,{2,7},15]

annularConsequences[8,{},16]

rectangularConsequences[8,{},16]+Plus@@(rectangularConsequences[8,{#},16]&/@Range[0,7])

annularConsequences[8,{},17]

rectangularConsequences[8,{},17]+Plus@@(rectangularConsequences[8,{#},17]&/@Range[0,7])

rectangularConsequences[8,{3,4},17]

rectangularConsequences[8,{2,5},17]

rectangularConsequences[8,{1,6},17]

rectangularConsequences[8,{0,7},17]

rectangularConsequences[8,{4,3,4},17]

annularConsequences[8,{},18]

rectangularConsequences[8,{},18]+Plus@@(rectangularConsequences[8,{#},18]&/@Range[0,7])

%%-%

rectangularConsequences[8,{0,7},18]

rectangularConsequences[8,{1,5},18]

rectangularConsequences[8,{0,6},18]

annularConsequences[8,{},19]

rectangularConsequences[8,{},19]+Plus@@(rectangularConsequences[8,{#},19]&/@Range[0,7])

%%-%

rectangularConsequences[8,{0,7},19]

%%-%

rectangularConsequences[8,{0,6},19]

rectangularConsequences[8,{0,5},19]

annularConsequences[8,{},20]

rectangularConsequences[8,{},20]+Plus@@(rectangularConsequences[8,{#},20]&/@Range[0,7])

rectangularConsequences[8,{},20]+Plus@@(rectangularConsequences[8,{#},20]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,#},20]&/@Range[0,7])

annularConsequences[8,{},21]

With[{k=21},rectangularConsequences[8,{},k]+Plus@@(rectangularConsequences[8,{#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,#},k]&/@Range[0,7])
]

annularConsequences[8,{},22]

With[{k=22},rectangularConsequences[8,{},k]+Plus@@(rectangularConsequences[8,{#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,#},k]&/@Range[0,7])
]

annularConsequences[8,{},25]

With[{k=25},rectangularConsequences[8,{},k]+Plus@@(rectangularConsequences[8,{#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,#},k]&/@Range[0,7])
]

annularConsequences[8,{},41]

With[{k=41},annularConsequences[8,{},k]-(rectangularConsequences[8,{},k]+Plus@@(rectangularConsequences[8,{#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,0,#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,0,0,#},k]&/@Range[0,7]))
]

FusionAtlasPaths={"~/projects/fusionatlas/code/package/","~/Documents/math papers/fusionatlas/code/package/","~/Documents/Math/fusionatlas/code/package/","~/Documents/fusionatlas/code/package"};
$Path=$Path~Join~FusionAtlasPaths;
<<FusionAtlas`

HaagerupBigraph

haagerupFamilyBigraph[7]

Names["Stable*"]

Table[DimensionOfLowWeightSpace[haagerupFamilyBigraph[7]][k],{k,0,20}]

PlanarAlgebraDimension[g_GradedBigraph][k_]:=PlanarAlgebraDimension[g][k]=MatrixPower[GraphAdjacencyMatrix[g],2k][[1,1]]

EH=haagerupFamilyBigraph[7];

PlanarAlgebraDimension[EH][10]

With[{k=10},rectangularConsequences[0,{},k]]

With[{k=10},rectangularConsequences[8,{},k]]

%%+%

With[{k=12},PlanarAlgebraDimension[EH][k]-(rectangularConsequences[0,{},k]+
rectangularConsequences[8,{},k]+
rectangularConsequences[8,{7,7},k]+
rectangularConsequences[8,{7,7,7},k]+Plus@@(rectangularConsequences[8,{#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,0,#},k]&/@Range[0,7])+Plus@@(rectangularConsequences[8,{0,0,0,#},k]&/@Range[0,7]))
]

rectangularConsequences[8,{7},10]

rectangularConsequences[8,{6,7},12]

rectangularConsequences[8,{6,7,7},12]

rectangularConsequences[8,{7,7,7,7},12]

rectangularConsequences[8,{0,6},12]

rectangularConsequences[8,{6},10]