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In Many Circles. Permutations as Products of Cycles.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
We have seen two different ways of looking at permutations. One considered permutations as linear orders of [n] and denoted them by specifying the order p1p2⋯pn of the n elements. The other considered permutations as elements of the symmetric group Sn and denoted them by parenthesized words that described the cycles of the permutations. We have not seen, however, too many connections between the two different lines of thinking. For instance, we have not analyzed the connection between the “visually” similar permutations 2417635 and (2)(41)(7635). Fortunately, such a connection exists, and it is a powerful tool in several enumeration problems. This is the content of the following well-known lemma that is due to Dominique Foata.
Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
The group of permutations of nletters (also called the symmetric group) is denoted as Sn. It is a finite group containing n!elements. The elements of Sncan be arranged in any order, and for any fixed arrangement we label the elements of Snas σi−1 for i = 1, …, n!We denote an arbitrary element σ ∈ Snas σ=(12…nσ(1)σ(2)…σ(n))
Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
All 11, onto functions f : S → S (permutations of S), where S is any non‐empty set, form a group under composition of functions. See Section 2.5.3. In particular, if S = {1, 2, 3, . . . , n}, the group of permutations of S is called the symmetric group, Sn. In Sn, each permutation can be written as a product of cycles. A cycle is a permutation σ = (i1i2...ik) $ (i_{1} i_{2} . . . i_{k} ) $ , where σ(i1) = i2, σ(i2) = i3, . . . , σ(ik) = i1. Each cycle of length greater than 1 can be written as a product of transpositions (cycles of length 2). A permutation is even (odd) if it can be written as the product of an even (odd) number of transpositions. (Every permutation is either even or odd.) The set of all even permutations in Sn is a normal subgroup, An, of Sn. The group An is called the alternating group on n elements.
Accurate screw detection method based on faster R-CNN and rotation edge similarity for automatic screw disassembly
Published in International Journal of Computer Integrated Manufacturing, 2021
Xinyu Li, Ming Li, Yongfei Wu, Daoxiang Zhou, Tianyu Liu, Fang Hao, Junhong Yue, Qiyue Ma
Finite group is a rather mature discipline in Mathematics, the properties and derived methods of which are widely used for image processing. The symmetric group defined over any set forms the finite group, the elements of which are all bijections from the set to itself, and the group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols, consists of permutation operations that can be performed on symbols. Because there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is . For regular polygons, the mapping invariance is mainly reflected in the equidistant transformations of their vertices, implying that they are entirely consistent, before and after a specific angle of rotation (Ledermann 1976).