In-tree depths of permutation-conjugation dynamics
Follow the latest progress by email.
Fix an n-cycle s and iterate D(q)=q s q⁻¹ on n-cycles. What is the greatest depth of a nonperiodic in-tree feeding a periodic orbit, and how does it depend on the prime-power factors of n?
Why it matters and what counts as progress
Why it matters: An exact depth formula explains transient behavior of this finite permutation dynamical system beyond enumeration of small cases.
What counts as progress: A proof of the odd-prime-power and power-of-two depth formulas and their maximum rule for general n, or a counterexample to one of them.
How the problem developed
- October 2026 · Manuscript release [1]
Frédéric Lefebvre-Naré A public PDF titled “October depth proof” was posted as a companion on in-tree depths of permutation-conjugation dynamics.
AI contribution
The in-tree of an n-cycle under cyclic conjugation: depth pᵉ⁻¹ for odd p, and 2ᵉ⁻¹−1 for p=2
For D(q)=q s q⁻¹ on n-cycles, Lefebvre-Naré’s October note claims depth p^(e−1) for odd prime powers p^e, depth 2^(e−1)−1 for powers of two, and the maximum prime-power depth for general n. The shorter proof uses a defect commutator and triangular Sylow-group filtration.
Sources
Website timeline
Publication dates and editorial changes.