Assert[JWCoeffFinder[{{},{}},n]==1]
Assert[JWCoeffFinder[{{1},{n}},n]==(1/(qInteger[n]))]
Assert[JWCoeffFinder[{{1,3},{1,3}},4]==(((1+qInteger[3]))/(qInteger[4]qInteger[3]qInteger[2]))]

JWCoeffFinder[{{},{}},n_]:=1

L is two lists, the first lists the positions of cups along the top edge, the second lists the position of caps along the bottom edge.

JWCoeffFinder[{cups_List,caps_List}, n_]/;(Length[cups]==Length[caps]):=
Module[{numLastCups,length,newCups,newCaps,innermostCaps},
length=Length[cups];
If[length==1,
numLastCups=1,
numLastCups=FindFirst[Reverse[Rest[cups]-Most[cups]],Except[1]]
];
If[Last[cups]+numLastCups<n,
innermostCaps=Select[caps,MemberQ[caps,#+1]==False&]~Join~{n},
innermostCaps=Select[caps,MemberQ[caps,#+1]==False&]];
newCups[i_]:=Take[cups,i-1]~Join~(Take[cups,{i+1,length}]-2);
If[Last[cups]+numLastCups<n,
newCaps[i_]:=Take[caps,i-1]~Join~(Take[caps,{i+1,length}]-2)~Join~{n},
newCaps[i_]:=Take[caps,i-1]~Join~(Take[caps,{i+1,length}]-2)];
Sum[((qInteger[innermostCaps[[i]]])/(qInteger[n]))*JWCoeffFinder[{newCups[i],newCaps[i]},n],{i,1,Length[innermostCaps]}]]

JWCoeffFinder[{{},{}},4]

JWCoeffFinder[{{1},{1}},3]

Clear[cups,caps,n,length,numLastCups,innermostCaps,newCups]

cups={1};
caps={1};
n=3;
length=Length[cups];
If[length==1,
numLastCups=1,
numLastCups=FindFirst[Reverse[Rest[cups]-Most[cups]],Except[1]]
]
If[Last[cups]+numLastCups<n,
innermostCaps=Select[caps,MemberQ[caps,#+1]==False&]~Join~{n},
innermostCaps=Select[caps,MemberQ[caps,#+1]==False&]]

newCups[i_]:=Take[cups,i-1]~Join~(Take[cups,{i+1,length}]-2)

If[Last[cups]+numLastCups<n,
newCaps[i_]:=Take[caps,i-1]~Join~(Take[caps,{i+1,length}]-2)~Join~{n},
newCaps[i_]:=
Take[caps,i-1]~Join~(Take[caps,{i+1,length}]-2)]

i=3
Take[cups,i-1]

Sum[((qInteger[innermostCaps[[i]]])/(qInteger[n]))*JWCoeffFinder[{newCups[i],newCaps[i]},n],{i,1,Length[innermostCaps]}]

L={{3},{3}}
Length[L[[2]]]
Length[L[[1]]]==Length[L[[2]]]
Clear[L]

MemberQ[{1,2,3},1]

L={2,3,7}
n=12
Select[L,MemberQ[L,#+1]==False&]
Position[Rest[L]-Most[L],1]
FindFirst[Reverse[Rest[L]-Most[L]],Except[1]]
Clear[L]

Position[Rest[L]-Most[L],#≠1&]