Necklaces, permutations, and periodic critical orbits for quadratic polynomials

Let Gₙ denote the nᵗʰ Gleason polynomial, whose roots correspond to parameters c such that the critical point $0$ is periodic of exact period n under iteration of z² + c, and let Ḡₙ denote the reduction of Gₙ modulo $2$. Buff, Floyd, Koch, and Parry made the surprising observation that the number of real roots of Gₙ is equal to the number of irreducible factors of Ḡₙ for all n. We provide a bijective proof for this result by first providing explicit bijections between (a) the set of real roots of Gₙ and the set N̄(n) of equivalence classes of primitive binary necklaces of length n under the inversion map swapping $0$ and $1$; and (b) the set of irreducible factors of Gₙ modulo 2 and the set Ñ⁺(n) of binary necklaces which are either primitive of length n with an even number of $1$'s or primitive of length $n/2$ with an odd number of $1$'s. We then provide an explicit bijection, closely related to Milnor and Thurston's kneading theory, between N̄(n) and Ñ⁺(n). In addition, we provide explicit bijections between N̄(n), the set CUP(n) of cyclic unimodal permutations of \ 1,…,n \, and the set N⁻(n) of primitive binary necklaces of length n with an odd number of $1$'s.