Transpositions, Simple Transpositions, Inversions and Sorting
A 2-cycle is known as a transposition. A simple transposition in Sn is a 2-cycle of the form (i i + 1).
For a permutation p in Sn, a pair (i, j)∈In is a permutation inversion, if when i<j, we have p(i) > p(j).
Every permutation can be written as a product of simple transpositions; furthermore, the number of simple transpositions one can write a permutation p in Sn can be the number of inversions of p and if the number of inversions in p is odd or even the number of transpositions in p will also be odd or even corresponding to the oddness of p.
Read more about this topic: Permutation Group
Famous quotes containing the word simple:
“You are all alike, you respectable people. You can’t tell me the bursting strain of a ten-inch gun, which is a very simple matter; but you all think you can tell me the bursting strain of a man under temptation. You daren’t handle high explosives; but you’re all ready to handle honesty and truth and justice and the whole duty of man, and kill one another at that game. What a country! What a world!”
—George Bernard Shaw (1856–1950)