|
In combinatorial mathematics, the necklace polynomials, or (Moreau's) necklace-counting function are the polynomials in ''α'' such that : By Möbius inversion they are given by : where is the classic Möbius function. The necklace polynomials are closely related to the functions studied by , though they are not quite the same: Moreau counted the number of necklaces, while necklace polynomials count the number of aperiodic necklaces. The necklace polynomials appear as: * the number of aperiodic necklaces (also called Lyndon words) that can be made by arranging beads the color of each of which is chosen from a list of colors (One respect in which the word "necklace" may be misleading is that if one picks such a necklace up off the table and turns it over, thus reversing the roles of clockwise and counterclockwise, one gets a different necklace, counted separately, unless the necklace is symmetric under such reflections.); * the dimension of the degree piece of the free Lie algebra on generators ("Witt's formula"); * the number of monic irreducible polynomials of degree over a finite field with elements (when is a prime power); * the exponent in the cyclotomic identity; *The number of Lyndon words of length ''n'' in an alphabet of size α.〔 ==Values== : :: where "gcd" is greatest common divisor and "lcm" is least common multiple. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Necklace polynomial」の詳細全文を読む スポンサード リンク
|