Permutation roots

This article discusses permutation roots, specifically square roots and cube roots, in the context of permutations on n elements. It defines a square root τ of a permutation σ such that applying τ twice equals σ. Using Python, it demonstrates composing permutations and counting square roots via brute force, noting the impracticality for large n due to factorial runtime. It presents a theorem stating a permutation has a square root iff the number of cycles of every even length is even. The article also touches on higher roots and the probability of random permutations having kth roots, referencing generating functions from Herbert Wilf's 'Generatingfunctionology'. This is a technical, math-focused piece relevant to computer science and algorithm theory.