Number of Permutations That Are mth Roots of Unity
This generalizes the concept of an involution. An mth root of unity is a permutation σ so that σm = 1 under permutation composition. Now every time we apply σ we move one step in parallel along all of its cycles. A cycle of length d applied d times produces the identity permutation on d elements (d fixed points) and d is the smallest value to do so. Hence m must be a multiple of all cycle sizes d, i.e. the only possible cycles are those whose length d is a divisor of m. It follows that the EGF g(x) of these permutations is
When m = p, where p is prime, this simplifies to
Read more about this topic: Random Permutation Statistics
Famous quotes containing the words number of, number, permutations, roots and/or unity:
“Even in ordinary speech we call a person unreasonable whose outlook is narrow, who is conscious of one thing only at a time, and who is consequently the prey of his own caprice, whilst we describe a person as reasonable whose outlook is comprehensive, who is capable of looking at more than one side of a question and of grasping a number of details as parts of a whole.”
—G. Dawes Hicks (18621941)
“I believe if we introduced the Lords Prayer here, senators would propose a large number of amendments to it.”
—Henry Wilson (18121875)
“Motherhood in all its guises and permutations is more art than science.”
—Melinda M. Marshall (20th century)
“April is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain.”
—T.S. (Thomas Stearns)
“The unity of effect or impression is a point of the greatest importance. It is clear, moreover, that this unity cannot be thoroughly preserved in productions whose perusal cannot be completed at one sitting.”
—Edgar Allan Poe (18091849)