kierrosluvut

Kierrosluvut are the lengths of the disjoint cycles in the cycle decomposition of a permutation of a finite set. For a permutation on n elements, the sum of the kierrosluvut equals n, and the collection of these lengths forms a partition of n. The order of the lengths is not important for the partition, but their multiplicities determine the permutation’s conjugacy class in the symmetric group.

Example: Consider the permutation of {1,2,3,4} given by σ(1)=2, σ(2)=3, σ(3)=1, σ(4)=4. Its cycle decomposition is (1 2 3)(4), so the kierrosluvut are 3 and 1.

Properties: Permutations with the same multiset of kierrosluvut are conjugate in S_n. If m_i denotes the number of cycles of length i, the number of permutations with that cycle type is n! / ∏_{i≥1} (m_i! i^{m_i}). The number of permutations of n with exactly k cycles is the unsigned Stirling number of the first kind, c(n,k). Thus kierrosluvut encode a partition of n and determine many combinatorial and algebraic aspects of the permutation.

Applications: Kierrosluvut appear in counting problems, in the study of permutation statistics, and in the representation theory of symmetric groups, where cycle structure determines conjugacy classes and influences character values. They provide a concise way to describe the qualitative shape of a permutation.