Maximum of the Stirling numbers

Question

It is known that the Stirling numbers of the first kind ${n\brack k}$ and second kind ${n\brace k}$ are unimodal. Is it known at which $k$'s the maximum occurs?

Answer 1

Stirling numbers of the first kind

In 1951, Hammersley [1] showed that the sequence of ${n\brack k}, k=1,\dots,n$ is unimodal, where the maximizing index $k_n$ is unique (later proved by Erdos [2]) and given by the following semi-closed-form formula (see (10) in [1]):

$$k_n=\left\lfloor\ln(n)+\gamma-1+\frac{\zeta(2)-\zeta(3)}{\ln(n)+\gamma-\frac32}+\frac{h(n)}{\left(\ln(n)+\gamma-\frac32\right)^2}\right\rfloor \\ =\left\lfloor\ln(n)+\gamma-1+o\left (\frac{1}{\ln(n)}\right)\right\rfloor $$

for some function $h(n)$ whose values vary in $(-1.098011, 1.430089)$. In this formula, $\gamma$ is the Euler-Mascheroni constant and $\zeta$ is the Riemann's zeta function. The above for $n > 189$ implies (see (4) in [2]) $$\left\lfloor\ln (n-1)+\frac12\right\rfloor\le k_n\le\lfloor\ln (n-1)\rfloor+1,$$

which means only one or two options should be checked to determine $k_n$ for $n>189$.

Carefully note that equations (5) and (6) in [1] and (4) in [2] are for the maximizing index of $\Pi_{n,k}$, i.e., the sum of the products of the first $n$ natural numbers taken $k$ at a time, for which we have (see here [3] for a proof) $$\Pi_{n,k}={n+1\brack n+1-k}.$$

Unfortunately, these equations have been mistakenly used for the maximizing index of ${n\brack k}$ (e.g., see the formula and bounds given for $k_n$ in this 2015 MSE answer [4] and this 2008 paper [5]).

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Stirling numbers of the second kind

The sequence of ${n\brace k}, k=1,\dots,n$ is unimodal, but the maximizing index $K_n$ was conjectured to be unique for $n\ge 3$ by Wegner in 1973 (still remained unsolved [6]), that is, the maximum value may be obtained for two consecutive indexes. Moreover, for $K_n$ no formula has been presented yet. It is known that $K_n \sim n/\ln n $ as $n\to \infty$ [7]. In 2009, Yu [8] proved the following bounds:

$$\left\lfloor e^{W(n)} \right\rfloor -2 \le K_n \le \left\lfloor e^{W(n)} \right\rfloor+1,$$

where $W$ denotes the Lambert-W function. As the length of the interval is always 3, one can determine $K_n$ by evaluating ${n\brace k}$ for only four values $k=\left\lfloor e^{W(n)} \right\rfloor+ i, i=-2,-1,0,1$.

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The maximum of r-Stirling numbers is studied in [5].

Answer 2

Let $X$ be the number of cycles in a random permutation of $n$ objects and let $Y$ be the number of parts in a random set partition of $n$ objects. You are asking about the modes of $X$ and $Y$. The mean of $X$ is known exactly, namely it is the harmonic number

$$\mathbb{E}(X) = H_n \approx \log n + \gamma.$$

In fact $X$ is the sum of $n$ independent Bernoulli random variables $B(1) + B \left( \frac{1}{2} \right) + \dots + B \left( \frac{1}{n} \right)$ so is asymptotically Gaussian with mean $H_n$, so the mode should be either the floor or ceiling of $\boxed{ H_n }$ (maybe off by $1$ in either direction at worst), and this could be checked experimentally.

The mean and mode of $Y$ are both approximately $\frac{n}{W(n)}$ where $W$ is the Lambert W function; see here for a discussion (without full details or error terms). The distribution of $Y$ is also asymptotically Gaussian. I think but am not sure that the mode is either the floor or the ceiling of $\boxed{ \frac{n}{W(n)} - 1 }$ (maybe off by $1$ in either direction at worst, again), and this could also be checked experimentally.

To put some concrete numbers to this let's take $n = 50$. Then

• For the Stirling cycle numbers, $H_{50} = 4.5 \dots$ so we should check $k = 4, 5$. We have $s(50, 4) = 7.1 \times 10^{63}$ and $s(50, 5) = 6.4 \times 10^{63}$ so the mode is $\boxed{ 4 = \lfloor H_{50} \rfloor }$ in this case.

• For the Stirling set / partition numbers, $\frac{50}{W(50)} - 1 = 16.5 \dots$ so we should check $k = 16, 17$. We have $S(50, 16) = 3.84 \dots \times 10^{46}$ and $S(50, 17) = 3.76 \dots \times 10^{46}$ so the mode is $\boxed{16 = \left\lfloor \frac{50}{W(50)} - 1 \right\rfloor }$ in this case.

The exact value of $k$ for the Stirling set numbers is A024417 on the OEIS. The sequence for the Stirling cycle numbers is a little hard to search for because it grows slowly.