#152
Latin Square Slicing
 

Difficulty:Hard
Topics:data-analysis math


A Latin square of order n is an n x n array that contains n different elements, each occurring exactly once in each row, and exactly once in each column. For example, among the following arrays only the first one forms a Latin square:

A B C    A B C    A B C
B C A    B C A    B D A
C A B    C A C    C A B

Let V be a vector of such vectors1 that they may differ in length2. We will say that an arrangement of vectors of V in consecutive rows is an alignment (of vectors) of V if the following conditions are satisfied:

  • All vectors of V are used.
  • Each row contains just one vector.
  • The order of V is preserved.
  • All vectors of maximal length are horizontally aligned each other.
  • If a vector is not of maximal length then all its elements are aligned with elements of some subvector of a vector of maximal length.
Let L denote a Latin square of order 2 or greater. We will say that L is included in V or that V includes L iff there exists an alignment of V such that contains a subsquare that is equal to L.

For example, if V equals [[1 2 3][2 3 1 2 1][3 1 2]] then there are nine alignments of V (brackets omitted):

 
        1              2              3

      1 2 3          1 2 3          1 2 3
  A   2 3 1 2 1    2 3 1 2 1    2 3 1 2 1
      3 1 2        3 1 2        3 1 2

      1 2 3          1 2 3          1 2 3
  B   2 3 1 2 1    2 3 1 2 1    2 3 1 2 1
        3 1 2        3 1 2        3 1 2

      1 2 3          1 2 3          1 2 3
  C   2 3 1 2 1    2 3 1 2 1    2 3 1 2 1
          3 1 2        3 1 2        3 1 2
Alignment A1 contains Latin square [[1 2 3][2 3 1][3 1 2]], alignments A2, A3, B1, B2, B3 contain no Latin squares, and alignments C1, C2, C3 contain [[2 1][1 2]]. Thus in this case V includes one Latin square of order 3 and one of order 2 which is included three times.

Our aim is to implement a function which accepts a vector of vectors V as an argument, and returns a map which keys and values are integers. Each key should be the order of a Latin square included in V, and its value a count of different Latin squares of that order included in V. If V does not include any Latin squares an empty map should be returned. In the previous example the correct output of such a function is {3 1, 2 1} and not {3 1, 2 3}.

1 Of course, we can consider sequences instead of vectors.
2 Length of a vector is the number of elements in the vector.


test not run
(= (__ '[[A B C D]
         [A C D B]
         [B A D C]
         [D C A B]])
   {})
test not run
(= (__ '[[A B C D E F]
         [B C D E F A]
         [C D E F A B]
         [D E F A B C]
         [E F A B C D]
         [F A B C D E]])
   {6 1})
test not run
(= (__ '[[A B C D]
         [B A D C]
         [D C B A]
         [C D A B]])
   {4 1, 2 4})
test not run
(= (__ '[[B D A C B]
         [D A B C A]
         [A B C A B]
         [B C A B C]
         [A D B C A]])
   {3 3})
test not run
(= (__ [  [2 4 6 3]
        [3 4 6 2]
          [6 2 4]  ])
   {})
test not run
(= (__ [[1]
        [1 2 1 2]
        [2 1 2 1]
        [1 2 1 2]
        []       ])
   {2 2})
test not run
(= (__ [[3 1 2]
        [1 2 3 1 3 4]
        [2 3 1 3]    ])
   {3 1, 2 2})
test not run
(= (__ [[8 6 7 3 2 5 1 4]
        [6 8 3 7]
        [7 3 8 6]
        [3 7 6 8 1 4 5 2]
              [1 8 5 2 4]
              [8 1 2 4 5]])
   {4 1, 3 1, 2 7})


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