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It is well-known that the number of surjections from a set of size n to a set of size m is quite a bit harder to calculate than the number of functions or the number of injections. (Of course, for surjections I assume that n is at least m and for injections that it is at most m.) It is also well-known that one can get a formula for the number of surjections using inclusion-exclusion, applied to the sets $X_1,...,X_m$, where for each $i$ the set $X_i$ is defined to be the set of functions that never take the value $i$. This gives rise to the following expression: $m^n-\binom m1(m-1)^n+\binom m2(m-2)^n-\binom m3(m-3)^n+\dots$.

Let us call this number $S(n,m)$. I'm wondering if anyone can tell me about the asymptotics of $S(n,m)$. A particular question I have is this: for (approximately) what value of $m$ is $S(n,m)$ maximized? It is a little exercise to check that there are more surjections to a set of size $n-1$ than there are to a set of size $n$. (To do it, one calculates $S(n,n-1)$ by exploiting the fact that every surjection must hit exactly one number twice and all the others once.) So the maximum is not attained at $m=1$ or $m=n$.

I'm assuming this is known, but a search on the web just seems to lead me to the exact formula. A reference would be great. A proof, or proof sketch, would be even better.

Update. I should have said that my real reason for being interested in the value of m for which S(n,m) is maximized (to use the notation of this post) or m!S(n,m) is maximized (to use the more conventional notation where S(n,m) stands for a Stirling number of the second kind) is that what I care about is the rough size of the sum. The sum is big enough that I think I'm probably not too concerned about a factor of n, so I was prepared to estimate the sum as lying between the maximum and n times the maximum.

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Whenever anyone has a question of the form "what is this function f:N-->N" then one very natural thing to do is to compute the first 10 values or so and then type it in to Sloane. research.att.com/~njas/sequences/index.html . If it's there then there will be references where you can read further. If it's not then this is evidence that little is known about it. I tried it with this sequence and got "ceiling of (n/sqrt(2))". Given that this is not the answer (e.g. f(1000)=722 and n/sqrt(2)=707.10678...) one might suspect that no exact formula is known but that n/sqrt(2) is close. –  Kevin Buzzard Jun 25 '10 at 12:23
    
More likely is that it's less than any fixed multiple of $n$ but by a slowly-growing amount, don't you think? –  JBL Jun 25 '10 at 12:27
    
These numbers also have a simple recurrence relation: mathoverflow.net/questions/27071/… –  JBL Jun 25 '10 at 12:28
    
@JBL: I have no idea what the answer to the maths question is. I just thought I'd advertise a general strategy, which arguably failed this time. –  Kevin Buzzard Jun 25 '10 at 12:30
    
See Herbert S. Wilf 'Generatingfunctionology', page 175. –  Bruce Arnold Jun 26 '10 at 21:32
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6 Answers

up vote 42 down vote accepted

It seems to be the case that the polynomial $P_n(x) =\sum_{m=1}^n m!S(n,m)x^m$ has only real zeros. (I know it is true that $\sum_{m=1}^n S(n,m)x^m$ has only real zeros.) If this is true, then the value of $m$ maximizing $m!S(n,m)$ is within 1 of $P'_n(1)/P_n(1)$ by a theorem of J. N. Darroch, Ann. Math. Stat. 35 (1964), 1317-1321. See also J. Pitman, J. Combinatorial Theory, Ser. A 77 (1997), 279-303. By standard combinatorics $$ \sum_{n\geq 0} P_n(x) \frac{t^n}{n!} = \frac{1}{1-x(e^t-1)}. $$ Hence $$ \sum_{n\geq 0} P_n(1)\frac{t^n}{n!} = \frac{1}{2-e^t} $$ $$ \sum_{n\geq 0} P'_n(1)\frac{t^n}{n!} = \frac{e^t-1}{(2-e^t)^2}. $$ Since these functions are meromorphic with smallest singularity at $t=\log 2$, it is routine to work out the asymptotics, though I have not bothered to do this.

Update. It is indeed true that $P_n(x)$ has real zeros. This is because $(x-1)^nP_n(1/(x-1))=A_n(x)/x$, where $A_n(x)$ is an Eulerian polynomial. It is known that $A_n(x)$ has only real zeros, and the operation $P_n(x) \to (x-1)^nP_n(1/(x-1))$ leaves invariant the property of having real zeros.

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Given that Tim ultimately only wants to sum m! S(n,m) rather than find its maximum, it is really only P_n(1) which one needs to compute. In principle this is an exercise in the saddle point method, though one which does require a nontrivial amount of effort. –  Terry Tao Jun 26 '10 at 19:03
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You don't need the saddle point method to find the asymptotic rate of growth of the coefficients of $1/(2−e^t)$. The smallest singularity is at $t=\log 2$. It is a simple pole with residue $−1/2$. Hence $$ P_n(1)\sim \frac{n!}{2(\log 2)^{n+1}}. $$ Using all the singularities $\log 2+2\pi ik, k\in\mathbb{Z}$, one obtains an asymptotic series for $P_n(1)$. It can be shown that this series actually converges to $P_n(1)$. –  Richard Stanley Jun 26 '10 at 19:51
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I quit being lazy and worked out the asymptotics for $P'_n(1)$. The Laurent expansion of $(e^t-1)/(2-e^t)^2$ about $t=\log 2$ begins $$ \frac{e^t-1}{(2-e^t)^2} = \frac{1}{4(t-\log 2)^2} + \frac{1}{4(t-\log 2)}+\cdots $$ $$ \qquad = \frac{1}{4(\log 2)^2\left(1-\frac{t}{\log 2}\right)^2} -\frac{1}{4(\log 2)\left(1-\frac{t}{\log 2}\right)}+\cdots, $$ whence $$ P'_n(1)= n!\left(\frac{n+1}{4(\log 2)^{n+2}}- \frac{1}{4(\log 2)^{n+1}}+\cdots\right). $$ Thus $P'_n(1)/P_n(1)\sim n/2(\log 2)$. –  Richard Stanley Jun 26 '10 at 21:00
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Ah, I didn't realise that it was so simple to read off asymptotics of a Taylor series from nearby singularities (though, in retrospect, I implicitly knew this in several contexts). Thanks, I learned something today! –  Terry Tao Jun 28 '10 at 20:26
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While we're on the subject, I'd like to recommend Flajolet and Sedgewick for anyone interested in such techniques: algo.inria.fr/flajolet/Publications/books.html –  Qiaochu Yuan Jun 29 '10 at 4:26
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This looks like the Stirling numbers of the second kind (up to the $m!$ factor).

This and this papers are specifically devoted to the maximal Striling numbers. It seems that for large $n$ the relevant asymptotic expansion is $$k! S(n,k)= (e^r-1)^k \frac{n!}{r^n}(2\pi k B)^{-1/2}\left(1-\frac{6r^2\theta^2 +6r\theta+1}{12re^r}+O(n^{-2})\right),$$ where $$e^r-1=k+\theta,\quad \theta=O(1),$$ $$B=\frac{re^{2r}-(r^2+r)e^r}{(e^r-1)^2}.$$

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If I understand correctly, what I (purely accidentally) called S(n,m) is m! times the Stirling number of the second kind with parameters n and m, which is conventionally denoted by S(n,m). So if I use the conventional notation, then my question becomes, how does one choose m in order to maximize m!S(n,m), where now S(n,m) is a Stirling number of the second kind? –  gowers Jun 25 '10 at 11:08
    
I've added a reference concerning the maximum Stirling numbers. For large $n$ $S(n,m)$ is maximized by $m=K_n\sim n/\ln n$. –  Andrey Rekalo Jun 25 '10 at 11:26
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Is it obvious how to get from there to the maximum of m!S(n,m)? –  gowers Jun 25 '10 at 11:29
    
Well, it's not obvious to me. But the computation for $S(n,m)$ seems to be not too complicated and probably can be adapted to deal with $m! S(n,m)$. –  Andrey Rekalo Jun 25 '10 at 11:36
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If f is an arbitrary surjection from N onto M, then we can think of f as partitioning N into m different groups, each group of inputs representing the same output point in M. The Stirling Numbers of the second kind count how many ways to partition an N element set into m groups. But this undercounts it, because any permutation of those m groups defines a different surjection but gets counted the same. There are m! such permutations, so our total number of surjections is

m! S(n,m)

To look at the maximum values, define a sequence S_n = n - M_n where M_n is the m that attains maximum value for a given n - in other words, S_n is the "distance from the right edge" for the maximum value. Computer-generated tables suggest that this function is constant for 3-4 values of n before increasing by 1. If this is true, then the m coordinate that maximizes m! S(n,m) is bounded by n - ceil(n/3) - 1 and n - floor(n/4) + 1.

I have no proof of the above, but it gives you a conjecture to work with in the meantime.

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The paper by Canfield and Pomerance that you quoted has an interesting expansion for $S(n,k+1)/S(n,k)$ at pag 5. The corresponding quotient $Q := Sur(n,k+1)/Sur(n,k)$ is just $k+1$ times as big; and sould be maximized by $k$ solving Q=1. –  Pietro Majer Jun 25 '10 at 14:16
    
ops sorry, the above comment was addressed to Andrey. –  Pietro Majer Jun 25 '10 at 21:15
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I found this paper of Temme (available here) that gives an explicit but somewhat complicated asymptotic for the Stirling number S(n,m) of the second kind, by the methods alluded to in previous answers (generating functions -> contour integral -> steepest descent)

Here's the asymptotic (as copied from that paper). One first sets

$t_0 := \frac{n-m}{m}$

and finds the positive real number $x_0$ solving the transcendental equation

$\frac{1-e^{-x_0}}{x_0} = \frac{m}{n}$

(one has the asymptotics $x_0 \approx 2(1-m/n)$ when $m/n$ is close to 1, and $x_0 \approx n/m$ when $m/n$ is close to zero.) One then defines

$A := \phi(x_0) - m t_0 + (n-m) t_0$

where

$\phi(x) := - n \ln x + m \ln(e^x - 1).$

(Note: $x_0$ is the stationary point of $\phi(x)$.) One has an integral representation

$S(n,m) = \frac{n!}{m!} \frac{1}{2\pi i} \int e^{\phi(x)} \frac{dx}{x}$

where the integral is a small contour around the origin. The saddle point method then gives

$S(n,m) = (1+o(1)) e^A m^{n-m} f(t_0) \binom{n}{m}$

where

$f(t_0) := \sqrt{\frac{t_0}{(1+t_0)(x_0-t_0)}}$

and o(1) goes to zero as $n \to \infty$ (uniformly in m, I believe).

In principle, one can now approximate $m! S(n,m)$ to within o(1) and compute its maximum in finite time, but this seems somewhat tedious. It does seem though that the maximum is attained when $m/n = c+o(1)$ for some explicit constant $0 < c < 1$.

EDIT: Actually, it's clear that the maximum is going to be obtained in the range $n/e \leq m \leq n$ asymptotically, because $m! S(n,m)$ equals $n! \approx (n/e)^n$ when $m=n$, and on the other hand we have the trivial upper bound $m! S(n,m) \leq m^n$. Among other things, this makes $x_0$ and $t_0$ bounded, and so the f(t_0) term is also bounded and not of major importance to the asymptotics. The other terms however are still exponential in n...

EDIT: There is also the identity

$\sum_{k=1}^n (k-1)! S(n,k) = (-1)^n Li_{1-n}(2)$

where $Li_s$ is the polylogarithm function. So, up to a factor of n, the question is the same as that of obtaining an asymptotic for $Li_{1-n}(2)$ as $n \to -\infty$. This seems quite doable (presumably from yet another contour integration and steepest descent method) but a quick search of the extant asymptotics didn't give this immediately.

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Richard's answer is short, slick, and complete, but I wanted to mention here that there is also a "real variable" approach that is consistent with that answer; it gives weaker bounds at the end, but also tells a bit more about the structure of the "typical" surjection. I'll write the argument in a somewhat informal "physicist" style, but I think it can be made rigorous without significant effort.

Tim's function $Sur(n,m) = m! S(n,m)$ obeys the easily verified recurrence $Sur(n,m) = m ( Sur(n-1,m) + Sur(n-1,m-1) )$, which on expansion becomes

$Sur(n,m) = \sum m_1 ... m_n = \sum \exp( \sum_{j=1}^n \log m_j )$

where the sum is over all paths $1=m_1 \leq m_2 \leq \ldots \leq m_n = m$ in which each $m_{i+1}$ is equal to either $m_i$ or $m_i+1$; one can interpret $m_i$ as being the size of the image of the first $i$ elements of $\{1,\ldots,n\}$. If we make the ansatz $m_j \approx n f(j/n)$ for some nice function $f: [0,1] \to {\bf R}^+$ with $f(0)=0$ and $0 \leq f'(t) \leq 1$ for all $t$, and use standard entropy calculations (Stirling's formula and Riemann sums, really), we obtain a contribution to $Sur(n,m)$ of the form

$\exp( n \int_0^1 \log(n f(t))\ dt + n \int_0^1 h(f'(t))\ dt + o(n) )$ (*)

where $h$ is the entropy function $h(\theta) := -\theta \log \theta - (1-\theta) \log (1-\theta)$. So, heuristically at least, the optimal profile comes from maximising the functional

$\int_0^1 \log(f(t)) + h(f'(t))\ dt$

subject to the boundary condition $f(0)=0$. (The fact that $h$ is concave will make this maximisation problem nice and elliptic, which makes it very likely that these heuristic arguments can be made rigorous.) The Euler-Lagrange equation for this problem is

$-\frac{f''}{f'(1-f')} = \frac{1}{f}$

while the free boundary at $t=1$ gives us the additional Neumann boundary condition $f'(1)=1/2$. The translation invariance of the Lagrangian gives rise to a conserved quantity; indeed, multiplying the Euler-Lagrange equation by $f'$ and integrating one gets

$\log(1-f') = \log f + C$

which is easily solved as

$f = \frac{1}{A} (1 - B e^{-At} )$

for some constants A, B. The Dirichlet boundary condition $f(0)=0$ gives $B=1$; the Neumann boundary condition $f'(1)=1/2$ gives $A=\log 2$, thus

$f(t) = (1 - 2^{-t}) / \log 2$.

In particular $f(1)=1/(2 \log 2)$, which matches Richard's answer that the maximum occurs when $m/n \approx 1/(2 \log 2)$. To match up with the asymptotic for $Sur(n,m)$ in Richard's answer (up to an error of $\exp(o(n))$, I need to have

$\int_0^1 \log f(t) + h(f'(t))\ dt = - 1 - \log \log 2.$

And happily, this turns out to be the case (after a mildly tedious computation.)

This calculation reveals more about the structure of a "typical" surjection from n elements to m elements for m free, other than that $m/n \approx 1/(2 \log 2)$; it shows that for any $0 < t < 1$, the image of the first $tn$ elements has cardinality about $f(t) n$. If one fixes $m$ rather than lets it be free, then one has a similar description of the surjection but one needs to adjust the A parameter (it has to solve the transcendental equation $(1-e^{-A})/A = m/n$).

With a bit more effort, this type of computation should also reveal the typical distribution of the preimages of the surjection, and suggest a random process that generates something that is within o(n) edits of a random surjection.

It's also interesting to note that the answer $m/n \approx 1/(2\log 2) = 0.72134\ldots$ fits extremely well with Kevin's numerical computation $f(1000)=722$, so we now have several independent confirmations that this is the correct answer...

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I found Terry's latest comment very interesting. Although his argument is not as easy as the complex variable technique and does not give the full asymptotic expansion, it is of much greater generality. It would make a nice expository paper (say for the Monthly) to present this example without any handwaving. –  Richard Stanley Jun 29 '10 at 18:59
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Hmm, not a bad suggestion. I may write a more detailed proof on my blog in the near future. –  Terry Tao Jun 30 '10 at 18:13
    
If I'm not wrong the asymptotics $m/n\sim 1/(2\log 2)$ is equivalent to $(m+1)^n\sim 4m^n$. Thus, for the maximal $m$ , the number of maps from $n$ to $m+1$ is approximatively 4 times the number of maps from $n$ to $m$ . I wonder if this may be proved by a direct combinatorial argument, yelding to another proof of the asymptotics. –  Pietro Majer Jul 2 '10 at 19:47
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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 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 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 asymptotic expansions for the maximal $m$ (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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OK this match quite well with the formula reported by Andrey Rekalo; the $r$ there is most likely coming from the stationary phase method. –  Pietro Majer Jun 25 '10 at 13:52
    
Pietro, I believe this is very close to how the asymptotic formula was obtained. –  Andrey Rekalo Jun 25 '10 at 14:03
    
yes, I think the starting point is standard and obliged. –  Pietro Majer Jun 25 '10 at 21:13
    
PS: Andrey, the papers you quoted initially where in pay-for journal, and led me to the wrong idea that there where no free version of that standard computation. My fault, I made a computation for nothing. –  Pietro Majer Jun 26 '10 at 6:24
    
Thank you for the comment. I'll try my best to quote free sources whenever I find them available. –  Andrey Rekalo Jun 26 '10 at 15:32
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