As is common terminology in graph reconstruction, given a graph $G$, we call a vertex deleted subgraph of $G$, a card, and call the multiset of all cards, the deck of $G$. The graph reconstruction problem is to determine the isomorphism type of $G$ by only looking at its deck.
Question: Is there a slick argument for reconstructing the number of Hamiltonian cycles of $G$ from its deck?
I have seen a paper of Tutte where he solves this using some machinery he uses to reconstruct several polynomial invariants of $G$. I don't have access to that paper right now, and I was wondering if a simpler solution exists when we restrict our attention just to counting Hamiltonian cycles.