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}.

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

(ns live.test
(:require [cljs.test :refer-macros [deftest is run-tests]]))

(defn squares [vecs]
(let [max-count (apply max (map count vecs))
positions #(range (inc (- max-count (count %))))
alignments (fn [v] (map #(concat
(repeat % nil)
v
(repeat (- max-count (count v) %) nil))
(positions v)))
cartesian-product (fn f [colls]
(if (empty? colls)
'(())
(for [x (first colls)
more (f (rest colls))]
(cons x more))))
all-planes (cartesian-product (map alignments vecs))
transpose #(apply map vector %)
get-slices-of (fn [n plane]
(map #(take n (drop % plane))
(range (inc (- (count plane) n)))))
get-squares-of (fn [n plane]
(map transpose (mapcat (comp (partial get-slices-of n) transpose)
(get-slices-of n plane))))
candidate-sizes (range 2 (inc (min (count vecs) max-count)))
all-candidates (mapcat (fn [n] (mapcat (partial get-squares-of n) all-planes))
candidate-sizes)
is-latin? (fn [square]
(and (every? (partial apply distinct?) square)
(every? (partial apply distinct?) (transpose square))
(= (count square) (count (set (flatten square))))
(not (contains? (set (flatten square)) nil))))]
(frequencies (map count (distinct (filter is-latin? all-candidates))))))

(deftest squares-test
(is (= (squares '[[A B C D]
[A C D B]
[B A D C]
[D C A B]])
{}))
(is (= (squares '[[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}))
(is (= (squares '[[A B C D]
[B A D C]
[D C B A]
[C D A B]])
{4 1, 2 4}))
(is (= (squares '[[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}))
(is (= (squares [  [2 4 6 3]
[3 4 6 2]
[6 2 4]  ])
{}))
(is (= (squares [[1]
[1 2 1 2]
[2 1 2 1]
[1 2 1 2]
[]       ])
{2 2}))
(is (= (squares [[3 1 2]
[1 2 3 1 3 4]
[2 3 1 3]    ])
{3 1, 2 2}))
(is (= (squares [[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})))

(run-tests)