  We show that the permanent of an $n\times n$ matrix of
$\operatorname{poly}(n)$-bit integers and the number of Hamiltonian cycles of
an $n$-vertex graph can both be computed in time $2^{n-\Omega(\sqrt{n})}$,
improving an earlier algorithm of Bj\"orklund, Kaski, and Williams
(Algorithmica 2019) that runs in time $2^{n - \Omega\left(\sqrt{n/\log \log
n}\right)}$.
  A key tool of our approach is to design a data structure that supports fast
"$r$-order evaluation" of permanent and Hamiltonian cycles, which cooperates
with the new approach on multivariate multipoint evaluation by Bhargava, Ghosh,
Guo, Kumar, and Umans (FOCS 2022).
