Chapter 4

Latin squares short range order and orthogonality

4.1. Short range coordination (clustering) in Latin squares of               order 4 and 5

Sites with equal values – equal numbers, same-coloured boards – appear as couples, triples, quadruples, strings, small triangles or squares (Figs 3.1.1, 3.2.1 and 3.6.1), or as singles. Equally valued sites will be considered singles at in-between distances of V5 and larger . Singles appear in Latin squares starting from order 5. In fact all sites are singles in LS’s in the order 5 family 5L (nr 14).

Equal-symbol nearest neighbours will be diagonally located with respect to each other at a distance of V2. Nearest neighbours numbers (nnn) are: 2 (at any position within a string and at the centre of a triple), 1 (couples, free ends of triples), or 0 (singles). Nearest neighbours counting at the edge of a LS will include nearest neighbours in the adjacent LS’s of the corresponding Latin-periodic structure to avoid break-off effects. We obtain coordination squares if we replace the values at all sites by the number nnn. We can define a clustering intensity value (clint) by taking a total count of nnn over the square and divide by n^2. It can be easily shown that that clint is invariant under point symmetry and translation symmetry movements of a LS, thus: the same for all members of a subfamily

We will quantify short range coordination in LS’s of order 4 and 5. Three different coordination squares (the first one with the cyclic shifting alternatives included) are obtained in the order 4 domain as well as three clint values ( below, with the corresponding families mentioned):

1   1   2   2                                   1   1   1   1                                                    2   2   2   2
1   1   2   2                                   1   1   1   1                                                    2   2   2   2
2   2   1   1                                   1   1   1   1                                                    2   2   2   2
2   2   1   1                                   1   1   1   1                                                    2   2   2   2

Family 2ST.2CC                   Family 4CC; types α and β                  Family 4ST; types α and β
LS’s 2 3 7 9 16 20 22 23      LS’s 4 6 8 10 11 12 13 14 15 17 19 21     LS’s 1 5 18 24
Clint 1.5                               Clint 1                                                   Clint 2

Some coordination results for different order 5 structure types are shown underneath with clint values. We find different coordination squares here for different structure types (α and β) within one family although the clint value is the same (34). Total counts vary from 34 to 16. Note that the total count is zero for family 5L (Table 3.2.1).

1   2   1   1    2     0   1   2   2   1      0   1   1   0  2      2  1  1  1  2    2  1  1  1  0     1   0 0 1   1     1  0  1   1   1
2  0   2   1    1      1   1   1   2   2      1   2   1   2   1      1  2  1  1  1     1  2  1  0  1     1   1  1  1  0     1  1  0  1  0
1   2   2   2   1      2   1   2   1   2      1   1   2   1   1      1  1  2  1  1     1  1  2  1   1     0  1  0  1  0     1 0  0  0  1
1   1   2   0   2      2   2   1   1   1      0   2   1   1   1      0 0  1  2  1     1  0  1  2  1     0  1  1  1   1     0  1  0  1   1
2  1    1   2    1      1   2   2   1   0     2   1   1   1    1      1  0  1  1  2    0  1  1  1  2     1  1  0  0  1      1   1  1  0  1
5α                           5β                         6α                       6β                  7                     12α                 12β
1.36                        1.36                       1.12                     1.12               1.12                 0.64             0.64

Conventional and colour  symmetry elements show up equally in coordination squares as illustrated in structure type nr.7: the main and back diagonal (above) correspond with an ordinary and a colour mirror line in the actual LS (table 3.2.2). We also note full fourfold symmetry in 12α and 12β representing colour fourfold axes in 12α and 12β actual, and two diagonal mirror lines in 5α and 5β, compared with the colour- elements indicated in Table 3.2.2. Symmetry of coordination square and corresponding connection diagram (par. 1.1) are similar.

For further refinement and more general use (orders 6 up to 9) it might be useful to add the next nearest neighbourships (at knights-move distance V5 in chess) to the area of short range coordination with appropriate coordination number( i.e. ½ )

All in all, coordination squares offer some directly obtainable structure- and symmetry viewpoints on lower order LS’s

4.2 Orthogonality

We have orthogonality if two LS’s when superposed have the property that each of the possible pairs of cell values occurs exactly once. This means that the contents ij of all combined cells are unique for LS’s in numeric format (values 1, 2, ..n). We will express orthogonal pairs with the constituting LS’s in standard format. The numbers of orthogonal pairs in general format will be n! x n! times larger since permutations leave the orthogonal relation intact.

Order 4: Ref. 1 and Ritter’s result (ref. 11) mention the number of (2x) 3456 orthogonal pairs (Ritter counts both AB and BA). Reduced to standard format: 6 pairs. Combining our 24 LS’s in 24×24 pairs we could confirm the following 6 orthogonal pairs :

(1, 14), (1, 21), (14, 21), ( 24, 4), (24, 11), (4, 11); numbers according to Fig. 2.1.1.

They can be collected in two mutually orthogonal triples (MOLS): (1, 14, 21) and (24, 4, 11).

Comparison with Table 2.4.1 and par. 3.5 shows that orthogonality in order 4 LS’s is allocated in – and restricted to – structure types 1 (4ST α) and 3 (4CC α); all in isotopy class (1) . All members, 1, 24 of 4STα and 4, 11, 14, 21 of 4CCα (par. 3.5) and thus: all members of class (1) – enter into (two) orthogonality relations. Relations (4 11) and (14 21 are within the 4CC family; the others are cross-overs between both families.

Self-orthogonality is orthogonality of a LS and its transpose . Graham and Roberts (ref. 12) report 48 self-orthogonal Latin squares (SOLS) in general format; this number reduces to 2 by leaving out he 4! permutations.

Two of the 24 order 4 LS’s are indeed self –orthogonal : nrs. 14 and 21; both of structure type 3 (4CC α). Note that both LS’s are diagonal (par. 2.1).

Order 5: Orthogonality literature mentions 3110400 (ref.13 ) as the number of orthogonal pairs for n=5, going down to 216 different pairs when the LS’s are in standard format.

We give three examples: (1) the mutually orthogonal quadruple:

1    2    3    4   5       1   2   3   4   5       1   2   3   4   5       1   2   3   4   5
2    3    4    5   1       5   1   2   3   4       3   4   5   1   2       4   5   1   2   3
3    4    5    1   2       4   5   1   2   3       5   1   2   3   4       2   3   4   5   1
4    5    1    2   3       3   4   5   1   2       2   3   4   5   1       5   1   2   3   4
5    1    2    3   4       2   3   4   5   1       4   5   1   2   3       3   4   5   1   2
1 (5ST)                     1 (5ST)                14 (5L)                14 (5L)

is often mentioned (ref. 9), resulting in 6 orthogonal pairs.

The structure types and families of the first and second two are readily identified as 1 (5ST) and 14 (5L) respectively (Table 3. 2. 1). The quadruple includes all members of each family (List 3. 2. 2). The two 5L-members are also self-orthogonal (leading to 2×5! = 240 SOLS in general format).

We also note (2) orthogonality of the two members selected to represent families ST. 4CCS and 4TC.L, structure types 7 and 9 ( List 3.2.2), checksums (par. 3.4) 7570 and 23928. This orthogonality holds if we apply each translation and/or dihedral group movement simultaneously on both selected members of types 7 and 9 (par. 3.3, Fig. 3.2.1). Fifty different pairs are involved because of the symmetry of both structure types (par.3.3). This ultimately yields 200 extra orthogonal pairs since each converted ST. 4TTC square shares orthogonality with two 4 TC.L members and vice versa.

This can be seen as follows: structure types 7 and 9 have family multiplicity 50. That means that if we prepare full multiplicity tables hk l for both of them we find each member four times. Let us select at random a LS of type 7, with index (path) 105, checksum19507 (par. 3.4). There will be three more paths leading to LS (19507). The four simultaniously produced LS’s of type 9 have cs’s 12823 and 7400, both twice. Again, there are two more paths for both of them leading to 12823 and 7400 in 9. Simultaneously in type 7 we have 4 paths leading to cs 9805 and ( and 4 with cs 19507). Altogether we have 8 orthogonal pairs, two by two the same, so the LS with cs 19507 is associated with four different pairs: 19507- 7400, 195 07-12823, 9805-7400, 9805- 12823. The same for any of the 50 LS. 200 orthogonal pairs.

(3). A similar “extended” orthogonality occurs with the selected member of 4TC.L and its only point symmetry counterpart ( 1 and 1cp respectively, see listing underneath). They are an orthogonal pair and the orthogonality holds if we if we move along the main diagonal of the corresponding Latin periodic structures, and try the successive 5×5 cuttings (“translation counterparts”) n and ncp (underneath) obtaining 5 orthogonal pairs.

1 2 3 4 5   1 2 3 4 5   1 2 3 4 5   1 2 3 4 5   1 2 3 4 5        1 2 3 4 5   1 2 3 4 5   1 2 3 4 5   1 2 3 4 5   1 2 3 4 5
4 5 2 3 1   3 5 4 2 1   4 3 1 5 2   5 3 1 2 4   5 4 2 1 3        5 3 4 1 2   5 4 1 3 2   3 5 2 1 4   3 4 2 5 1   4 3 2 5 1
5 3 4 1 2   4 1 2 5 3   2 4 5 3 1   2 5 4 1 3   4 3 5 2 1        4 5 2 3 1   2 3 5 1 4   5 1 4 2 3   4 1 5 3 2   5 4 2 1 3
3 1 5 2 4   5 4 1 3 2   5 1 4 2 3   3 4 2 5 1   2 5 1 3 4        2 4 1 5 3   3 5 4 2 1   2 4 5 3 1   5 3 1 2 4   3 1 4 5 2
2 4 1 5 3   2 3 5 1 4   3 5 2 1 4   4 1 5 3 2   3 1 4 5 2        3 1 5 2 4   4 1 2 5 3   4 3 1 5 2   2 5 1 4 3   2 5 1 3 4
4TC.L    1             2              3              4              5               1cp           2cp           3cp          4cp          5cp
transl. 0,0            1,1           2,2           3,3          4,4             0,0            1,1             2,2           3,3           4,4

The 4TC.L Latin squares 1… 5 and 1cp…. 5cp , last example, are also self – orthogonal. These extra SOLS are given above in standard numeric format The total number of SOLS for order 5 would thus rise to12x5! (1440), in numerical agreement with ref. 8.

Taking (1), (2), (3) together the number of order five orthogonal pairs obtained in standard format is 211, slightly below the literature total: 216. Try to find the others!

We do note again that all orthogonal pairs and self orthogonals obtained are located in a single isotopy class, class1) (par. 3.3); in four structure families 5ST, 5L, ST.4CCS, 4TC.L .