Arrange the table cells according to the number of elements in the cell and add a column containing the number of elements in each cell

Question

Arrange the table cells according to the number of elements in the cell and add a column containing the number of elements in each cell.

If I have the lists

S1 = {Subscript[x, 1] Subscript[x, 2] , Subscript[x, 3] };
S2 = {Subscript[x, 4] Subscript[x, 5], 
   Subscript[x, 6] Subscript[x, 7] Subscript[x, 8]};

I use this code to create a table

TableForm[
 ToExpression /@ (StringRiffle[#, {"{", "", "}"}] & /@ 
    Tuples[{S1, S2}]), TableHeadings -> Automatic]

The output is in the form:

I need to sort the table cells according to the number of items in the cell in ascending order.

Then add a column containing the number of elements in each cell. The output will be in the form

Thanks for the help.

Answer 1

Convert each tuple to a simple list so the elements can be counted. Make a table of the lists multiplied together, with the length of each list. Sort the table by length.

S1 = {Subscript[x, 1] Subscript[x, 2] , Subscript[x, 3]};
S2 = {Subscript[x, 4] Subscript[x, 5], 
   Subscript[x, 6] Subscript[x, 7] Subscript[x, 8]};

TableForm[
  SortBy[
    Table[{Times[Sequence@@t], Length@t},
      {t, Flatten/@Tuples[{S1, S2} /. Times -> List]}],
    Last],
  TableHeadings -> {Automatic, {"cut element", "order"}}]

Here's a simpler method that uses Variables to make lists from the tuples.

TableForm[
  SortBy[
    Table[{Times[Sequence@@t], Length@t},
     {t, Variables/@Tuples[{S1, S2}]}],
    Last],
  TableHeadings -> {Automatic, {"cut element", "order"}}]