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:
“‘Tis the gift to be simple ‘tis the gift to be free
‘Tis the gift to come down where you ought to be
And when we find ourselves in the place just right
‘Twill be in the valley of love and delight.”
—Unknown. ‘Tis the Gift to Be Simple.
AH. American Hymns Old and New, Vols. I–II. Vol. I, with music; Vol. II, notes on the hymns and biographies of the authors and composers. Albert Christ-Janer, Charles W. Hughes, and Carleton Sprague Smith, eds. (1980)