= Solution
A rate-$\lambda$ <Poisson process> has $N_0=0$, right-continuous paths with upward jumps of size one, stationary <independent> increments, and $N_t-N_s\sim\operatorname{Poisson}(\lambda(t-s))$ for $t\geq s$. Assume $\lambda>0$. Its successive waiting times are <independent> exponential variables of rate $\lambda$. For $0<t_1<\cdots<t_n<t$, the product of their densities and the <probability> of no further jump before $t$ is
$$
\lambda^ne^{-\lambda t_1}e^{-\lambda(t_2-t_1)}\cdots e^{-\lambda(t_n-t_{n-1})}e^{-\lambda(t-t_n)}=\lambda^ne^{-\lambda t}.
$$
Divide by $\mathbb P(N_t=n)=e^{-\lambda t}(\lambda t)^n/n!$ to obtain the conditional joint density
$$
\boxed{\frac{n!}{t^n}\mathbf1_{\{0<t_1<\cdots<t_n<t\}}.}
$$
Boundary inequalities do not affect a density. Equivalently these are the ordered values of $n$ <independent> uniform points of $[0,t]$. For $n=0$, the conditional tuple is empty.
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