[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the number of partial permutations is given by

[[math]] |\widetilde{S}_N|=\sum_{k=0}^Nk!\binom{N}{k}^2 [[/math]]

that is, [math]1,2,7,34,209,\ldots\,[/math], and that we have the estimate

[[math]] |\widetilde{S}_N|\simeq N!\sqrt{\frac{\exp(4\sqrt{N}-1)}{4\pi\sqrt{N}}} [[/math]]

in the [math]N\to\infty[/math] limit.