Super-polynomial Chow rank of the additive listing for cycle-graph isomorphism

Prove that the Chow rank (product/split rank) of the polynomial P_{~C_n}(A) = ∑_{σ ∈ S_n/Aut(C_n)} ∏_{(i,j) ∈ E(σ C_n σ^{-1})} a_{i,j}, where a_{i,j} are adjacency indeterminates and C_n denotes the n-vertex cycle graph, grows super-polynomially with n.

Background

Within the differential computer framework, P_{~G}(A) is the additive listing polynomial that sums monomial edge listings over all graphs isomorphic to a fixed graph G. For G = C_n (the n-vertex cycle), evaluating P_{~C_n} on the adjacency matrix B_H of a graph H counts the number of Hamiltonian cycles in H, a #P-complete quantity.

Chow decompositions of P_{~C_n} correspond to depth-three (ΣΠΣ) arithmetic formulas computing P_{~C_n}. Establishing a super-polynomial lower bound on the Chow rank of P_{~C_n} would therefore yield strong lower bounds in low-depth arithmetic circuit complexity for this natural family of polynomials tied to Hamiltonian cycle counting.