Listen bigwigs say so long Pólya theorem, finally have time to learn about the theorem.
Replacement (permutation) is simply a (full) arrangement. For example, \ (1,2,3,4 \) is a replacement for the \ (3,1,2,4 \) . In general, we write \ (I \) to \ (a_i (1 <= i <= n) \) is replaced with a
\[ \left ( \begin{matrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{matrix} \right ) \]
Obviously, the nature of displacement is one mapping, so we can be above replacement abbreviated as \ (f = \ {A_1, A_2, \ cdots, A_N \} \) . Wherein \ (F (I) = a_i (. 1 <= I <= n-) \) .
Replacement can be complex. If \ (f = \ {A_1, A_2, \ cdots, A_N \}, G = \ {B_1, B_2, \ cdots, B_n \} \) , we call \ (fg = \ {b_ { a_1}, b_ { a_2}, \ cdots, b_ { a_n} \} \) of \ (F \) and \ (G \) compound. It means that we first of a number \ (i \) mapped to \ (f (i) \) , and then mapped to \ (G (f (i)) \) . For example, \ (F = \ {1,3,4,2 \}, G = \ {3,2,1,4 \} \) , then \ (fg = \ {3,1,4,2 \ } \) , which represents \ (2 \) to be mapped to \ (F (2) = 3 \) , the \ (3 \) and then mapped to the \ (G (3) =. 1 \) , so in general, \ (FG (2) = G (F (2)) =. 1 \) .