0. ,
The separator ",".
1.
{}
The minimum container is one dimensional and empty.
Rule-1: {,} = {}
Sameness in width is not allowed.
The separator is Nothingness.
Rule-2: {{}} = {}
Sameness in depth is not allowed.
The set collapses into one ordered set;
[]
2.
{,{}}
Rule-3: {,{}} = {{},}
Order is not allowed.
The first iteration of the set in depth and width expands into 2 ordered sets;
[,[]] + [[],]
3.a.
{,{,{}}}
The second iteration in depth expands into 4 ordered sets;
[,[,[]]] + [,[[],]] + [[,[]],] + [[[],],]
The number of the ordered sets of the pure depth expansion is 2**(n-1).
3.b.
{,{},{,{}}}
The second iteration in width collapses into 12 ordered sets;
[,[],[,[]]] + [,[,[]],[]] +
[[],,[,[]]] + [[],[,[]],] +
[[,[]],[],] + [[,[]],,[]] +
[,[],[[],]] + [,[[],],[]] +
[[],,[[],]] + [[],[[],],] +
[[[],],[],] + [[[],],,[]]
The number of the ordered sets of the pure width expansion is sf(n) = 3!*2!*1!*0!=12 where sf is Superfactorial. It is also known as Barnes G-function.
3.c.
Two hybrid sets emerge at level-3 between pure depth and pure width sets;
3H1={0,2} = {,{,{}}}
3H2={1,2} = {{},{,{}}}
which expand into 4 ordered sets each;
[,[,[]]] + [,[[],]] + [[,[]],] + [[[],],]
[[],[,[]]] + [[],[[],]] + [[,[]],[]] + [[[],],[]]
Thus level-3 has 4 unordered sets and 4+12+4+4=24 ordered sets.
4.a. {,{,{,{}}}}
The third iteration in depth expands into 8 ordered sets;
[,[,[,[]]]] + [,[,[[],]]] + [,[[,[]],]] + [,[[[],],]] +
[[[[],],],] + [[[,[]],],] + [[,[[],]],] + [[,[,[]]],]
4.b. {, {}, {,{}}, {,{},{,{}}}}
The third iteration in width collapses into sf(n)=4!*3!*2!*1!*0!=576 ordered sets.
4.c.
The hybrid sets at level-4 are calculated from the available set;
{0,1,2,3D,3W,3H1,3H2}
by taking all combinations that produce new sets for level-4. For example {0,3D} gives 4D which already exists. The number of sets that contain at least one of {3D,3W,3H1,3H2} is 120. For example a few of them;
{1,3D} = {{}, {,{,{}}}}
{0,3W} = {, {,{},{,{}}}}
{3D,3W} = {{,{,{}}}, {,{},{,{}}}}
{1,2,3D,3W} = {{}, {,{}}, {,{,{}}}, {,{},{,{}}}}
{3H1,3H2} = {{,{,{}}}, {{},{,{}}}}
Each expand into many ordered sets giving a large number.
...
It is obvious that the number of sets will explode from here on. The number of the ordered sets is much more. This method will find all possible sets and expand into infinity very fast. The addressing of each set can also be calculated.
