rotmönsteruppsättning

Rotmönsteruppsättning is a Swedish term that refers to a collection of rotation patterns used to describe cyclic transformations in mathematical structures or linguistic constructs. The word is composed of rot (root) or rotation, mönster (pattern) and uppsättning (set). In mathematics it usually denotes a set of cycles that represent the action of a permutation group on a finite set. In cryptography it may describe a group of simple Caesar‑type shift ciphers, each represented by a fixed displacement of characters. Within Swedish mathematics education it is often introduced as part of the study of permutation groups, illustrating how complex permutations can be decomposed into disjoint cycles.

The concept originated in the early 20th century, when Swedish mathematicians developed group‑theoretic methods for analyzing symmetry in combinatorial problems. The resulting terminology was published in the *Svensk Matematisk Tidskrift* before being adopted in university curricula. Since its introduction, rotmönsteruppsättning has been used in courses on abstract algebra, combinatorics and computer science for illustrating orbit structures, stabilisers, and Burnside's lemma. It also finds application in puzzle design, where the rotational symmetry of a configuration can be catalogued by a rotmönsteruppsättning.

Examples include the rotation set { (1 2 3), (1 3 2) } that describes the two non‑trivial 3‑cycles of the symmetric group S₃, and the cipher set { ROT3, ROT5, ROT7 } common in Swedish cryptographic tutorials. Studies on the enumeration of cycle types in symmetric groups often refer to the total number of distinct rotmönsteruppsättningar as a function of the set size.

References are available in standard Swedish mathematics handbooks and on the Wikipedia page for permutation cycles.