Permutative automorphisms of the Cuntz algebras: Quadratic cycles, an involution and a box product

Permutative automorphisms of the Cuntz algebra O n are in bijection with a class permutations k elements, that called stable, and further partitioned by rank. In this work we mainly focus on stable cycles quadratic case (i.e., = 2 ). More precisely, such provide characterization rank one (so proving Conjecture 12.1 [3] ), exhibit closed formula for number r -cycles (valid all characterize enumerate 3-cycles any given We also show set is equipped natural involution preserves cycle-type rank, there map associates to two m respectively, permutation ( ) elements.