How to do permutations9/5/2023 ![]() ![]() What happens when there's only one character 'c'? There's only one permutation of that element, and so we return a list containing only that element. Recursively, think about the base case and build from that intuition. The Working with Permutations tutorial was introduced in Maple 2015.įor more information on Maple 2015 changes, see Updates in Maple 2015. Thus, a cycle type of 2, 2, 3, 5, 5 indicates a permutation that is a product of two transpositions, one 3 -cycle and two 5 -cycles. , c k, where the lengths of the cycles in a disjoint cycle decomposition of the permutation are listed in non-decreasing order. The multi-set of lengths of these cycles is called the "cycle type" of the permutation, and the PermCycleType command computes the cycle type as a list of the form c 1, c 2. The PermParity command computes this homomorphism.Įvery permutation can be written, in an essentially unique way, as a product of disjoint cycles. This defines a homomorphism from the symmetric group S n to the multiplicative group −1, 1. The "parity" of p is defined to be 1 if p can be written as a product of an even number of transpositions, and is defined to be −1 otherwise. There are infinitely many such products for any given permutation, but the number of transpositions in such a product is either always even or always odd. b of two permutations a and b is computed by the PermCommutator command.Įvery permutation p can be written as a product of transpositions.To compute the conjugate a b of a permutation a by a permutation b, you can use either exponential notation, or the PermConjugate command. The PermOrder command computes the order of a permutation. The smallest positive integer n for which a n is the identity permutation is called the "order" of the permutation a. Note that computing powers of a permutation is efficient for large powers.Īlternatively, you can use the PermPower command. You can also compute the inverse of a permutation by using the PermInverse command. In particular, you can compute the inverse of a permutation as a power with −1. You can compute a power a n, for an integer n, of a permutation a, by using the `^` operator. a are different permutations that is, multiplication of permutation is, in general, ' not' commutative.is used to multiply (or compose) two permutations.Ī := Perm( ] ) b := Perm( ] ) The noncommutative multiplication operator. The PermFixed command returns the set of those points less or equal to the degree of a permutation that are fixed by it. The largest point moved by a permutation is called the "degree" of the permutation, you can compute it by using the PermDegree command. To compute the set of moved points, use the PermSupport command.ġ, 3, 4, 6, 7 ![]() The set of points that are actually displaced by a permutation is called the "support" of the permutation. You can also use the PermApply command to compute the image of a point, or to map a permutation onto a set or list of points. This is important to understand when multiplying (or composing) permutations. Note: Despite the notation, permutations act on the ' right'. To apply a permutation p to a point i, you use the "indexed" notation p. Many of the commands available for computing with permutations reside in the GroupTheory package, and the following command can be used to access them. Note that permutations are displayed in disjoint cycle notation. In this earlier example, the sublist represents the cycle (transposition) that interchanges the points 1 and 3, and the sublist represents the cycle that takes 2 to 5, 5 to 4 and 4 to 2. To create a permutation by specifying its disjoint cycle structure, use nested lists in which each sublist represents the corresponding cycle. In the first case, you use a list L of the form, where a_i is the image of i under the permutation. To create a permutation in Maple, you must specify either an explicit list of the images of the integers in the range 1.n, or the disjoint cycle structure of the permutation. ![]()
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |