How to group the table in the following way?

Question

Consider the following table:

tabb = {{1, 2, 3, 4}, {5, 1, 7, 8}, {1, 2, 3, 4, 13, 16}, {5, 6, 7, 9}, {1, 2, 3, 4, 11, 16}, {5, 6, 7, 8}, {1, 2, 3, 4, 11, 16, 13, 35}, {1, 1, 3, 4, 11, 16, 13, 35}, {1, 1, 14, 8, 33, 8, 13, 35}};

Could you please tell me how to group it by the number of columns and also the product of each second column?

I.e., the rows {{1, 2, 3, 4}, {5, 1, 7, 8}, {5, 6, 7, 9},{5, 6, 7, 8}} of tabb have the same number of columns, but the products of the 2nd and 4th columns are different. Therefore, they should be considered as different elements: the first two rows form one element, while the remaining two form another two elements.

I probably should use GroupGy[tabb,{Length,condition}] where condition specifies the product of the columns. My problem is that I cannot generalize the condition to an arbitrary number of columns.

Answer 1

GroupBy takes as the second argument an arbitrary function which can serve as a "grouping parameter". As you have correctly identified it, you can use a list {length, product} as a parameter. To get every second element of a list, use Span.

GroupBy[tabb, {Length[#], Times @@ #[[2 ;; ;; 2]]} &]

(* <| {4, 8} -> {{1, 2, 3, 4}, {5, 1, 7, 8}}, 
      {6, 128} -> {{1, 2, 3, 4, 13, 16}, {1, 2, 3, 4, 11, 16}}, 
      {4, 54} -> {{5, 6, 7, 9}}, 
      {4, 48} -> {{5, 6, 7, 8}}, 
      {8, 4480} -> {{1, 2, 3, 4, 11, 16, 13, 35}}, 
      {8, 2240} -> {{1, 1, 3, 4, 11, 16, 13, 35}, {1, 1, 14, 8, 33, 8, 13, 35}}|> *)