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Some Comments on Philatelic Latin Squares from Pakistan

Some Comments on Philatelic Latin Squares from Pakistan

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438 ong>Someong> ong>Commentsong> on Philatelic Latin Squares from Pakistan A 5 × 5 PLS was issued by Pakistan (Figure 4.1) in 2004 for the National Philatelic Exhibition in Lahore on Universal Postal Union Day (October 9). This PLS depicts five kinds of tropical fish: neon tetra, striped gourami, black widow, yellow dwarf cichlid, and tiger barb. The “neon tetra” (Paracheirodon innesi) is a freshwater fish of the characin family (family Characidae) of order Characiformes. The type species of its genus, it is native to blackwater or clearwater streams in southeastern Colombia, eastern Peru, and western Brazil. The giant gourami (Osphronemus goramy) is a freshwater fish belonging to the family Osphronemidae, also sometimes known as the banded gourami, rainbow gourami, or “striped gourami”, native to India. The black tetra (Gymnocorymbus ternetzi), also known as the black skirt tetra or “black widow” tetra, is a freshwater fish of the characin family (family Characidae), of order Characiformes and is native to the Paraguay and Guaporé River basins of southern Brazil, Argentina, and Bolivia. The “yellow dwarf cichlid” (Apistogramma borellii) is found in shallow swampy regions along rivers in South America. The “tiger barb” (Puntius tetrazona) is a species of tropical freshwater fish belonging to the Puntius genus of the minnow family. The natural geographic range reportedly extends throughout the Malay peninsula, Sumatra and Borneo, with unsubstantiated sightings reported in Cambodia [85]. The Latin square in Figure 4.1 has several interesting properties. In particular it is of the type known as a “knight’s move” or “Knut Vik” design, as described by Preece [62]: Of Latin squares used for crop experiments with a single set of treatments, the earliest examples (published in 1924) are 5 × 5 squares of the systematic type known as Knut Vik [84] or “knight’s move” designs (Knut Vik being a [Norwegian] person, not a Scandinavian translation of “knight’s move”!); these are squares where all cells containing any one of the treatments can be visited by a succession of knight’s moves (as in chess) and where no two diagonally adjacent cells have the same treatment. In 1931 Tedin [78] and in 1951 Nissen [56] observe that there are two possible arrangements of the 5 × 5 Knut Vik design in standard-form: A,B,C,D,E in sequence in the top row. The arrangement KV1 [78, Fig. 1, Arr. 2] has treatment A (in row 2) below C (in row 1) and KV2 [78, Fig. 1, Arr. 1] has A (in row 2) below D (in row 1), see (4.1): ⎛ ⎞ ⎛ ⎞ A ⎜ ⎜D KV1 = ⎜ ⎜B ⎜ ⎝E B E C A C A D B D B E C E ⎟ C ⎟ A ⎟ ⎟, ⎟ D⎠ A ⎜ ⎜C KV2 = ⎜ ⎜E ⎜ ⎝B B D A C C E B D D A C E E ⎟ B ⎟ D ⎟ ⎟. ⎟ A⎠ (4.1) C D E A B D E A B C

Ka Lok Chu et al. 439 The Latin square KV1 is a two-steps forwards circulant while KV2 is a two-steps backwards circulant. Yates [88], following Tedin [78], notes that KV1 and KV2 are special balanced 5 × 5 Latin squares in which the treatments are as evenly spaced as possible. A strict knight’s move square is a Latin square such that every two cells which contain the same symbol are joined by a knight’s path composed of cells which contain that symbol. Two cells, not in the same row or column, are said to be diagonally adjacent when their distance apart is 2. As Owens [57] has established, there exists a strict knight’s move square of order n without diagonal adjacencies if and only if n = 5. To within permutations of the symbols and transposition there is only one such square: KV1 or KV2 in (4.1). The first publication of KV1 or KV2 that we have found is that of KV1 over 300 years ago by Poignard [59, fig. XXIV, p. 74 (1704)]. A few years earlier in 1691, La Loubère9 [50,51] considered the Latin squares ⎛ ⎞ ⎛ ⎞ c e b d a 0 χ ω φ ψ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜a c e b d⎟ ⎜χ ω φ ψ 0⎟ ⎜ ⎟ ⎜ ⎟ GL1 = ⎜ ⎜d a c e b ⎟ ⎟, GL2 = ⎜ ⎜ω φ ψ 0 χ ⎟ ⎟, (4.2) ⎜ ⎟ ⎜ ⎟ ⎜ ⎝b d a c e⎟ ⎜ ⎠ ⎝φ ψ 0 χ ω⎟ ⎠ e b d a c ψ 0 χ ω φ which we call La Loubère matrices [77, (3.1)]. The Latin square GL1 is a one-step forwards circulant while GL2 is a one-step backwards circulant. Moreover, GL1 and GL2 are orthogonal to each other in that every ordered pair of symbols from GL1 and GL2 occurs exactly once, and so the pairs (GL1, GL2) and (G2, G1) form Graeco-Latin squares. We believe that the pair (GL2, GL1) considered by La Loubère (1691) is the first Graeco- Latin square to use Greek and Latin letters [77, §3] preceding the seminal work [34, 35] by Leonhard Euler (1707–1783). The PLS in Figure 4.1 is of type KV2, which is a two-steps backwards circulant; transposing KV2 and recoding the treatments yields KV1 (in standard-form), which is a twosteps forwards circulant as noted by Fisher [41, (book 2) p. 78]: In this arrangement the areas bearing each treatment are nicely distributed over the experimental area, so as to exclude all probability that the more important components of heterogeneity should influence the comparison between treatments. 9 Simon de La Loubère (1642–1729), the French poet and special envoy of King Louis XIV (1638–1715) to Narai the Great (1629–1688), King of the Ayutthaya kingdom of Siam.

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