Packing and counting arbitrary Hamilton cycles in random digraphs

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in $\mathcal{D}(n,p)$ for nearly optimal $p$ (up to a ${\log }^{c}n$ factor). In particular, we show that given $t=(1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$,eachofwhichisorientedarbitrarily,adigraph$D \sim {\cal D}(n,p)$w\mathrm{.}h\mathrm{.}p\mathrm{.}containsedgedisjointcopiesof$C_1,\ldots ,C_t$,provided$p=\omega(\log ^3 n/n)$\mathrm{.}Wealsoshowthatgivenanarbitrarilyoriented$n$-vertexcycle$C$,arandomdigraph$D \sim {\cal D}(n,p)$w\mathrm{.}h\mathrm{.}p\mathrm{.}contains$(1\pm o(1))n!p^n$copiesof$C$,provided$p \geq \log ^{1 + o(1)}n/n$.