In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column.
Dalam kombinatorika dan rancangan percobaan, persegi Latin adalah persegi berukuran n × n yang berisi n simbol berbeda, masing-masing hanya muncul sekali pada setiap kolom dan pada setiap baris.
An example of a 3 × 3 Latin square is The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols, but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3.
Contoh dari persegi Latin 3×3 adalah Istilah "persegi Latin" terinspirasi oleh makalah matematika oleh Leonhard Euler (1707–1783), yang menggunakan alfabet Latin sebagai simbol.
The Korean mathematician Choi Seok-jeong was the first to publish an example of Latin squares of order nine, in order to construct a magic square in 1700, predating Leonhard Euler by 67 years.
Matematikawan Korea Choi Seok-jeong pertama kali menerbitkan sebuah contoh persegi Latin berukuran 9×9, dalam upayanya membuat persegi ajaib pada tahun 1700; lebih awal 67 daripada Leonhard Euler.
For example, the Latin square above is not reduced because its first column is A, C, B rather than A, B, C. Any Latin square can be reduced by permuting (that is, reordering) the rows and columns.
Sebagai contoh, persegi Latin pada contoh di atas tidak tereduksi karena kolom pertamanya adalah A, C, B, bukan A, B, C. Semua persegi Latin dapat direduksi dengan mempermutasi (yakni, menyusun ulang) urutan kolom dan baris.
For example, the orthogonal array representation of the Latin square is { (1, 1, 1), (1, 2, 2), (1, 3, 3), (2, 1, 2), (2, 2, 3), (2, 3, 1), (3, 1, 3), (3, 2, 1), (3, 3, 2) }, where for example the triple (2, 3, 1) means that in row 2 and column 3 there is the symbol 1.
Sebagai contoh, representasi larik ortogonal dari persegi Latin adalah {(1, 1, 1), (1, 2, 2), (1, 3, 3), (2, 1, 2), (2, 2, 3), (2, 3, 1), (3, 1, 3), (3, 2, 1), (3, 3, 2) }.
One classic result is that ∏ k1 n ( k ! ) n / k ≥ L n ≥ ( n ! ) 2 n n n 2 .
Salah satu hasil menyatakan bahwa ∏ k1 n ( k ! ) n / k ≥ L n ≥ ( n ! ) 2 n n n 2 .
This formula for the number Ln of n × n Latin squares is L nn !
Rumus ini menyatakan banyaknya persegi Latin berukuran n × n adalah L nn !
∑ A ∈ B n ( − 1 ) σ 0 ( A ) ( per ⁡ A n ) ,n!\sum _{A\in B_{n}}^{}(-1)^{\sigma _{0}(A)}{\binom {\operatorname {per} A}{n}},} where Bn is the set of all n × n {0, 1}-matrices, σ0(A) is the number of zero entries in matrix A, and per(A) is the permanent of matrix A. The table below contains all known exact values.
∑ A ∈ B n ( − 1 ) σ 0 ( A ) ( per ⁡ A n ) ,n!\sum _{\mathbf {A} \in B_{n}}^{}(-1)^{\sigma _{0}(\mathbf {A} )}{\binom {\operatorname {per} \mathbf {A} }{n}},} dengan Bn menyatakan himpunan semua matriks binear berukuran n × n, σ 0 ( A ) (\mathbf {A} )} menyatakan banyak elemen nol di matriks A } , dan per ⁡ A \mathbf {A} } adalah permanen dari matriks A } .
I, II (Second ed.).
I, II (Edisi Second).
Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Second ed.).
Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Edisi Second).
Design and Analysis of Experiments, Volume 2: Advanced Experimental Design (First ed.).
Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Edisi Second).
In S. Ghosh and C. R. Rao (ed.).
Dalam S. Ghosh and C. R. Rao (ed.).
Amsterdam: North-Holland Publishing Co. pp.
Amsterdam: North-Holland Publishing Co. hlm.
Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.).
Constructions and Combinatorial Problems in Design of Experiments (Edisi corrected reprint of the 1971 Wiley).
Experimental Design with Applications in Management, Engineering, and the Sciences (2nd edition (November 28, 2017) ed.).
Experimental Design with Applications in Management, Engineering, and the Sciences (Edisi 2nd edition (November 28, 2017)).
Weisstein, Eric W. "Latin Square".
(Inggris) Weisstein, Eric W. "Latin Square".