Permutations, tensor products, and Cuntz algebra automorphisms

We study the reduced Weyl groups of the Cuntz algebras O-n from a combinatorial point of view. Their elements correspond bijectively to certain permutations of n(r) elements, which we call stable. We mostly focus on the case r = 2 and general n. A notion of rank is introduced, which is subadditive in a suitable sense. Being of rank 1 corresponds to solving an equation which is reminiscent of the Yang-Baxter equation. Symmetries of stable permutations are also investigated, along with an immersion procedure that allows to obtain stable permutations of (n + 1)(2) objects starting from stable permutations of n(2) objects. A complete description of stable transpositions and of stable 3-cycles of rank 1 is obtained, leading to closed formulas for their number. Other enumerative results are also presented which yield lower and upper bounds for the number of stable permutations.