OEIS A000315: Reduced Latin Squares

Join us as we explore A000315, the counts of reduced Latin squares — n-by-n grids filled with n symbols where the first row and first column are in natural order. Reduction removes symmetries so counting becomes feasible, yielding the sequence 1, 1, 1, 4, 56, 9408, ... with no simple closed form. The classic formula (published in 1992) involves matrix permanents and heavy computation. A striking Stones–Wanless 2010 result then ties the combinatorics to number theory: An is divisible by n for composite n, and An ≡ 1 (mod n) for prime n. Beyond counting, Latin squares illuminate quasi-groups/loops, experimental design in statistics, Sudoku-type puzzles, and coding theory via orthogonal pairs. It’s a vivid example of how a simple grid touches algebra, combinatorics, and information theory.

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