Cuntz algebra automorphisms: Graphs and stability of permutations

We characterize the permutative automorphisms of the Cuntz algebra O n (namely, stable permutations) in terms of two sequences of graphs that we associate to any permutation of a discrete hypercube [n](t). As applications we show that in the limit of large t (resp. n) almost all permutations are not stable, thus proving Conj. 12.5 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], characterize (and enumerate) stable quadratic 4 and 5-cycles, as well as a notable class of stable quadratic r-cycles, i.e. those admitting a compatible cyclic factorization by stable transpositions. Some of our results use new combinatorial concepts that may be of independent interest.