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I don't have a precise reference for your problem (given $n$ find "the most surjected" $m$); waiting for a precise one, I can say that I think the standard starting point should be as follows. To avoid confusion I modify slightly your notation for the surjection surjections from an $n$ elements set to an $m$ elements set into $\mathrm{Sur}(n,m).$ One has the generating function (coming e.g. from the analogous g.f. for Stirling numbers of second kind)

$$(e^x-1)^m\,=\sum_{n\ge m}\ \mathrm{Sur}(n,m)\ \frac{x^n}{n!}\ ,$$

whence by the Cauchy formula with a simple integration contour around 0 ,

$$\frac{\mathrm{Sur}(n,m)}{n!}={1 \over 2\pi i} \oint \frac{(e^z-1)^m}{z^{n+1}}dz$$

For a circular path $re^{it}$ we find

$$\frac{\mathrm{Sur}(n,m)}{n!}={1 \over 2\pi } \int_{-\pi}^{\pi}\left(\exp(re^{it})-1\right)^m e^{-int} dt\ .$$

This holds for any number $r>0$, and a the most convenient one should be chosen according to the stationary phase method; here a change of variable followed by dominated convergence may possibly give a convergent integral, producing an asymptotics: this is e.g. how one can derive the Stirling asymptotics for n!.

In your case, the problem is: for a given $n$ (large) maximize the integral in $m$, and give an asymptotic expansion expansions for the maximal $m$. m$ (the first order should be $\lambda n + O(1)$ with $ 2/3\leq \lambda\leq 3/4 $ according to Michael Burge's exploration). This seems to be tractable; for the moment I leave this few hints hoping they are useful, but I'm very curious to see the final answer.
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I don't have a precise reference for your problem (given $n$ find "the most surjected" $m$); waiting for a precise one, I can say that I think the standard starting point should be as follows. To avoid confusion I modify slightly your notation into $\mathrm{Sur}(n,m).$ For for the surjection from an $n$ elements set to an $m$ elements set , one into $\mathrm{Sur}(n,m).$ One has the generating function (coming e.g. from the analogous g.f. for Stirling numbers of second kind)

$$(e^x-1)^m\,=\sum_{n\ge m}\ \mathrm{Sur}(n,m)\ \frac{x^n}{n!}\ ,$$

whence by the Cauchy formula with a simple integration contour around 0 ,

$$\frac{\mathrm{Sur}(n,m)}{n!}={1 \over 2\pi i} \oint \frac{(e^z-1)^m}{z^{n+1}}dz$$

For a circular path $re^{it}$ we find

$$\frac{\mathrm{Sur}(n,m)}{n!}={1 \over 2\pi } \int_{-\pi}^{\pi}\left(\exp(re^{it})-1\right)^m e^{-int} dt\ .$$

This holds for any number $r>0$, and a most convenient one should be chosen according to the stationary phase method(; here a change of variable followed by dominated convergence may possibly give a convergent integral, producing an asymptotics: this is e.g. how one can derive the Stirling asymptotics for n!)n!.

In your case, the problem is, : for a given $n$ (large) maximize the integral in $m$, and give an asymptotic expansion for the maximal $m$. This seems to be tractable; for the moment I leave this few hints hoping they are useful, but I'm very curious to see the final answer.
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I don't have a precise reference for your problem (given $n$ find "the most surjected" $m$); waiting for a precise one, I can say that I think the standard starting point should be as follows. To avoid confusion I modify slightly your notation into $\mathrm{Sur}(n,m).$ For the surjection from an $n$ elements set to an $m$ elements set, one has the generating function (coming e.g. from the analogous g.f. for Stirling numbers of second kind)

$$(e^x-1)^m\,=\sum_{n\ge m}\ \mathrm{Sur}(n,m)\ \frac{x^n}{n!}\ ,$$

whence by the Cauchy formula with a simple integration contour around 0 ,

$$\frac{\mathrm{Sur}(n,m)}{n!}={1 \over 2\pi i} \oint \frac{(e^z-1)^m}{z^{n+1}}dz$$

For a circular path $re^{it}$ we find

$$\frac{\mathrm{Sur}(n,m)}{n!}={1 \over 2\pi } \int_{-\pi}^{\pi}\left(\exp(re^{it})-1\right)^m e^{-int} dt\ .$$

This holds for any number $r>0$, and a most convenient one should be chosen according to the stationary phase method (here a change of variable followed by dominated convergence may possibly give a convergent integral, producing an asymptotics: this is e.g. how one can derive the Stirling asymptotics for n!).

In your case, the problem is, for a given $n$ (large) maximize the integral in $m$, and give an asymptotic expansion for the maximal $m$. This seems to be tractable; for the moment I leave this few hints hoping they are useful, but I'm very curious to see the final answer.