Components and Cycles of Random Mappings

1 Stepanov

For introductory purposes, let us examine solely one-to-one mappings ${1, 2, \dots, n} \to {1, 2, \dots, n}$ , i.e., permutations on $n$ symbols. Let $Λ$ denote the length of the longest cycle in an $n$ -permutation, chosen uniformly at random. We have

\begin{matrix} lim n \to \infty P {Λ \leq a n} = ρ (\frac{1}{a}), & 0 < a \leq 1 \end{matrix}

where, from the DDE,

ρ (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} 1 - ln (x) & if 1 < x \leq 2, 1 - \frac{π^{2}}{12} - ln (x) + \frac{1}{2} ln (x)^{2} + {Li}_{2} (\frac{1}{x}) & if 2 < x \leq 3. \end{matrix}

Thus, for example,

	$lim n \to \infty P {\frac{1}{3} < \frac{Λ}{n} \leq \frac{1}{2}}$	$= ρ (2) - ρ (3)$
		$= \frac{π^{2}}{12} - ln (2) + ln (3) - \frac{1}{2} ln (3)^{2} - {Li}_{2} (\frac{1}{3})$
		$= 0.258244431148....$

Let us now explore a less-familiar approach [11]. Define a function $h (ξ)$ to be equal to $0$ for $ξ < 1$ and $1 / ξ$ for $ξ \geq 1$ . Using the series expansion for $exp (- E (η))$ in terms of $E (η)$ , we have

L^{- 1} [\frac{exp (- E (η))}{η}] = \infty \sum k = 0 \frac{(- 1)^{k}}{k!} L^{- 1} [\frac{E (η)^{k}}{η}]

where the power $E (η)^{k}$ is the Laplace transform of the $k$ -fold self-convolution $h_{k} (ξ)$ of $h (ξ)$ . On the one hand, formulas

are well-known. On the other hand,

L^{- 1} [\frac{E (η)^{2}}{η}] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} l l l - \frac{π^{2}}{6} + ln (ξ)^{2} + 2 {Li}_{2} (\frac{1}{ξ}) & if ξ \geq 2, 0 & if 0 \leq ξ < 2. \end{matrix}

does not appear in [12]. Since $h_{k} (ξ) = 0$ for $0 \leq ξ < k$ , only terms $k = 0, 1, \dots, ⌊ ξ ⌋$ of the series need to be summed. Consequently, for $1 / 3 < a \leq 1 / 2$ ,

L^{- 1} {[\frac{exp (- E (η))}{η}]}_{ξ \to 1 / a} = 1 - ln (\frac{1}{a}) + \frac{1}{2} [- \frac{π^{2}}{6} + ln (a)^{2} + 2 {Li}_{2} (a)]

which is consistent with before.

2 Mutafchiev

Let us now remove the restriction that mappings be one-to-one. Let $Λ$ denote the length of the largest component in an $n$ -mapping, chosen uniformly at random. We have

\begin{matrix} lim n \to \infty P {Λ \leq a n} = \frac{1}{\sqrt{a}} σ (\frac{1}{a}), & 0 < a \leq 1 \end{matrix}

where, from the DDE,

\begin{matrix} \sqrt{x} σ (x) = 1 - \frac{1}{2} ln ⎛ ⎜ ⎜ ⎝ \frac{1 + \sqrt{} 1 - \frac{1}{x}}{1 - \sqrt{1 - \frac{1}{x}}} ⎞ ⎟ ⎟ ⎠ & if 1 < x \leq 2. \end{matrix}

For reasons of simplicity, change our example domain from $[1 / 3, 1 / 2]$ to $[1 / 2, 2 / 3]$ . Hence

	$lim n \to \infty P {\frac{1}{2} < \frac{Λ}{n} \leq \frac{2}{3}}$	$= \sqrt{\frac{3}{2}} σ (\frac{3}{2}) - \sqrt{2} σ (2)$
		$= - \frac{1}{2} ln ⎛ ⎜ ⎜ ⎝ \frac{1 + \sqrt{1 - \frac{2}{3}}}{1 - \sqrt{1 - \frac{2}{3}}} ⎞ ⎟ ⎟ ⎠ + \frac{1}{2} ln ⎛ ⎜ ⎜ ⎝ \frac{1 + \sqrt{1 - \frac{1}{2}}}{1 - \sqrt{1 - \frac{1}{2}}} ⎞ ⎟ ⎟ ⎠$
		$= 0.222894638557....$

Let us again explore the less-familiar approach [13]. Define $h (ξ)$ as previously. From

\sqrt{π ξ} L^{- 1} ⎡ ⎢ ⎢ ⎣ \frac{exp (- \frac{1}{2} E (η))}{\sqrt{η}} ⎤ ⎥ ⎥ ⎦ = \sqrt{π ξ} \infty \sum k = 0 \frac{(- 1)^{k}}{2^{k} k!} L^{- 1} [\frac{E (η)^{k}}{\sqrt{η}}]

we recognize two well-known formulas:

\begin{matrix} L^{- 1} [\frac{1}{\sqrt{η}}] = \frac{1}{\sqrt{π ξ}}, & L^{- 1} [\frac{E (η)}{\sqrt{η}}] = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{2}{\sqrt{π ξ}} a r c t a n h (\sqrt{1 - \frac{1}{ξ}}) & if ξ \geq 1, 0 & if 0 \leq ξ < 1. \end{matrix} \end{matrix}

Since $h_{k} (ξ) = 0$ for $0 \leq ξ < k$ , only terms $k = 0, 1, \dots, ⌊ ξ ⌋$ of the series need to be summed. Consequently, for $1 / 2 < a \leq 1$ ,

\sqrt{\frac{π}{a}} L^{- 1} {⎡ ⎢ ⎢ ⎣ \frac{exp (- \frac{1}{2} E (η))}{\sqrt{η}} ⎤ ⎥ ⎥ ⎦}_{ξ \to 1 / a} = 1 - a r c t a n h (\sqrt{1 - a}) = 1 - \frac{1}{2} ln (\frac{1 + \sqrt{1 - a}}{1 - \sqrt{1 - a}})

which again is consistent with before.

As an aside, had we kept the domain as $[1 / 3, 1 / 2]$ , then numerics are possible:

	$lim n \to \infty P {\frac{1}{3} < \frac{Λ}{n} \leq \frac{1}{2}}$	$= 1 / 2 \int 1 / 3 \frac{1}{2 t^{3 / 2}} σ (\frac{1 - t}{t}) d t$
		$= \frac{1}{2} 1 / 2 \int 1 / 3 \frac{1}{t \sqrt{1 - t}} ⎡ ⎢ ⎢ ⎣ 1 - \frac{1}{2} ln ⎛ ⎜ ⎜ ⎝ \frac{1 + \sqrt{1 - \frac{t}{t - 1}}}{1 - \sqrt{1 - \frac{t}{t - 1}}} ⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ d t$
		$= 0.110414874191....$

but symbolics seem unlikely. A closed-form expression for the inverse Laplace transform of $E (η)^{2} / \sqrt{η}$ also remains open.

3 Purdom & Williams

We change the subject to cycles, i.e., loops in the functional graph. Let $Λ$ denote the length of the longest cycle in an $n$ -mapping, chosen uniformly at random. Our goal is to find ${lim}_{n \to \infty} P {Λ \leq b \sqrt{n}}$ . While no relevant DDE is yet known, there is an associated inverse Laplace transform

due to Mutafchiev [13]. Unlike previously, $0 < b < \infty$ holds (rather than $0 < a < 1$ ) and $E (\sqrt{2 b η})$ is the Laplace transform of $\frac{1}{2 ξ} erfc (\sqrt{\frac{b}{2 ξ}})$ . Self-convolutions of this function do not enjoy the same vanishing properties as those for $h (ξ)$ . Truncating the infinite series, although it is convergent, will unfortunately lead to non-zero error. An alternative expression

\sqrt{2 π} \frac{1}{2 π i} 1 + i \infty \int 1 - i \infty exp (- E (b ζ) + \frac{ζ^{2}}{2}) d ζ

is due to Stepanov [11]. Letting $η = \frac{b ζ^{2}}{2}$ , i.e., $ζ = \sqrt{\frac{2 η}{b}}$ and $d ζ = \frac{d η}{\sqrt{2 b η}}$ , demonstrates that this complex contour integral is equal to the preceding inverse Laplace transform.

There is, thankfully, a different approach available. If the number $N$ of cyclic points in a random mapping is fixed, then as Kolchin [14] wrote, “… these $N$ cyclic points… form a random permutation”. This suggests multiplying [15]

the conditional probability of

Λ

given

N

P {Λ \leq λ \sqrt{n} | N = ν \sqrt{n}} = ρ (\frac{ν}{λ})

by [16, 17] the limiting density (as $n \to \infty$ ) of $N$ :

\begin{matrix} ν exp (\frac{- ν^{2}}{2}) & (Rayleigh)% \end{matrix}

and then differentiating (with respect to $λ$ ) to obtain the joint density of $(Λ, N)$ :

	$f (λ, ν)$	$= ν exp (\frac{- ν^{2}}{2}) ρ^{'} (\frac{ν}{λ}) (\frac{- ν}{λ^{2}})$
		$= ν exp (\frac{- ν^{2}}{2}) ρ (\frac{ν}{λ} - 1) (\frac{- λ}{ν}) (\frac{- ν}{λ^{2}})$
		$= \frac{ν}{λ} exp (\frac{- ν^{2}}{2}) ρ (\frac{ν - λ}{λ})$

where $0 < λ < ν < \infty$ . We have not seen this formula in the literature: it is apparently new. For $r \geq 2$ , let $Λ_{r}$ denote the length of the $r^{th}$ longest cycle in an $n$ -mapping, chosen uniformly at random. If the permutation has no $r^{th}$ cycle, then its $r^{th}$ longest cycle is defined to have length $0$ . Define the $r^{th}$ generalized Dickman function $ρ_{r} (ξ)$ to satisfy

\begin{matrix} ξ ρ_{r}^{'} (ξ) + ρ_{r} (ξ - 1) = ρ_{r - 1} (ξ - 1) for ξ > 1, & ρ_{r} (ξ) = 1 for 0 \leq ξ \leq 1 \end{matrix}

where $ρ_{1} = ρ$ . By the same argument, for $r \geq 2$ ,

Define

G_{r, h} = \frac{1}{h! (r - 1)!} \infty \int 0 x^{h - 1} E (x)^{r - 1} exp [- E (x) - x] d x

(in this paper, rank $r = 1, 2, 3$ or $4$ ; height $h = 1$ or $2$ ). Purdom & Williams [18], building on Shepp & Lloyd [19]

, discovered asymptotic formulas for moments

E (Λ_{r}^{h})

. We can easily verify their findings:

lim n \to \infty \frac{E (Λ_{r})}{\sqrt{n}} = \sqrt{\frac{π}{2}} G_{r, 1} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 0.782481600991656615 01... & if r = 1, 0.26267067265131265469... & if r = 2, 0.11068781528281010827... & if r = 3, 0.05056118481134243184... & if r = 4; \end{matrix}

lim n \to \infty \frac{V (Λ_{r})}{n} = 2 G_{r, 2} - \frac{π}{2} G_{r, 1}^{2} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 0.24111407342881901748.. . & if r = 1, 0.04395998473216610374... & if r = 2, 0.01233552055537805858... & if r = 3, 0.00386619224804518754... & if r = 4. \end{matrix}

The mode of $Λ_{1}$ occurs when

	$0$	$= \frac{d}{d λ} \infty \int λ f (λ, ν) d ν = - f (λ, λ) + \infty \int λ \frac{\partial f}{\partial λ} (λ, ν) d ν$
		$= - exp (\frac{- λ^{2}}{2}) + \infty \int λ ν exp (\frac{- ν^{2}}{2}) \frac{\partial}{\partial λ} [\frac{1}{λ} ρ (\frac{ν}{λ} - 1)] d ν;$

the inner derivative becomes

	$\frac{- 1}{λ^{2}} ρ (\frac{ν}{λ} - 1) + \frac{1}{λ} ρ^{'} (\frac{ν}{λ} - 1) (\frac{- ν}{λ^{2}})$	$= \frac{- 1}{λ^{2}} ρ (\frac{ν}{λ} - 1) - \frac{1}{λ} \frac{ρ (\frac{ν}{λ} - 2)}{\frac{ν}{λ} - 1} (\frac{- ν}{λ^{2}})$
		$= \frac{- 1}{λ^{2}} ρ (\frac{ν - λ}{λ}) + \frac{ν}{λ^{2} (ν - λ)} ρ (\frac{ν - 2 λ}{λ});$

solving the equation

exp (\frac{- λ^{2}}{2}) = \frac{1}{λ^{2}} \infty \int λ [- ρ (\frac{ν - λ}{λ}) + \frac{ν}{ν - λ} ρ (\frac{ν - 2 λ}{λ})] ν exp (\frac{- ν^{2}}{2}) d ν

yields $0.4809...$ as the mode. The median $0.6842...$ of $Λ_{1}$ arises simply from

\frac{1}{2} = \infty \int λ \infty \int μ f (μ, ν) d ν d μ .

For $Λ_{2}$ , the mode is $0$ ; we did not pursue the median. Another new asymptotic result is the cross-correlation between $Λ_{r}$ and $N$ :

lim n \to \infty \frac{E (Λ_{r} N) - E (Λ_{r}) E (N)}{\sqrt{V (Λ_{r})} \sqrt{V (N)}} = \frac{\sqrt{2 - \frac{π}{2}} G_{r, 1}}{\sqrt{2 G_{r, 2} - \frac{π}{2} G_{r, 1}^{2}}} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} c c c 0.83298010... & if r = 1, 0.65486924... & if r = 2, 0.52094617... & if r = 3, 0.42505712... & if r = 4. \end{matrix}

Using formulas in [10, 20, 21], it is possible to similarly compute the cross-correlation between $Λ_{r}$ and $Λ_{s}$ where $r < s$ .

4 Rényi

A mapping is said to be connected (or indecomposable) if it possesses exactly one component. This is a rare event, in the sense that

\begin{matrix} P {M = 1} \sim \sqrt{\frac{π}{2 n}} & as n \to \infty \end{matrix}

where $M$ counts the components. Let $Λ$ denote the length of the unique cycle in a connected $n$ -mapping, chosen uniformly at random. Our goal is to find ${lim}_{n \to \infty} P {Λ \leq b \sqrt{n}}$ as before, but the circumstances are vastly simpler. Rényi [22] proved that the limiting density (as $n \to \infty$ ) of $Λ$ is

\begin{matrix} \sqrt{\frac{2}{π}} exp (\frac{- λ^{2}}{2}) & (half-normal) \end{matrix}

for $0 < λ < \infty$ , which implies immediately that

lim n \to \infty \frac{E (Λ)}{\sqrt{n}} = \sqrt{} \frac{2}{π} = 0.79788456080286535587...,

lim n \to \infty \frac{V (Λ)}{n} = 1 - \frac{2}{π} = 0.36338022763241865692....

It is surprising that arbitrary mappings and connected mappings differ so little here ( $0.7824...$ versus $0.7978...$ ). We might have expected that uniqueness would carry more influence.

Of course, $N = Λ$ when there is just one component. Allowing instead $m$ components, where $m \geq 2$ is a fixed integer, rarity persists [23, 24]

\begin{matrix} P {M = m} \sim \frac{1}{2^{m - 1} (m - 1)!} \sqrt{\frac{π}{2 n}} ln (n)^{m - 1} & as n \to \infty \end{matrix}

but the asymptotic values $\sqrt{2 / π}$ and $1 - 2 / π$ for $E (N)$ and $V (N)$ evidently do not change. Section 6 contains more on this issue.

5 Pavlov

For arbitrary mappings, the expected number of components [25, 26] is $\sim \frac{1}{2} ln (n)$ . If our constraint from Section 4 loosens so that $m \to \infty$ but so slowly that $m / ln (n) \to 0$ , then Rényi’s formula still applies, as proved by Pavlov [24]. This leads us to a set of conjectural results comparable to those in Section 3.

Let $Λ$ be the longest cycle length and $N$ be the cyclic points total. As before, a conditional probability coupled with the limiting density (as $n \to \infty$ ) of $N$ :

\sqrt{\frac{2}{π}} exp (\frac{- ν^{2}}{2})

suffice to give the joint density of $(Λ, N)$ :

f (λ, ν) = \sqrt{\frac{2}{π}} \frac{1}{λ} exp (\frac{- ν^{2}}{2}) ρ (\frac{ν - λ}{λ})

where $0 < λ < ν < \infty$ . Further,

f_{r} (λ, ν) = \sqrt{\frac{2}{π}} \frac{1}{λ} exp (\frac{- ν^{2}}{2}) [ρ_{r} (\frac{ν - λ}{λ}) - ρ_{r - 1} (\frac{ν - λ}{λ})]

for $r \geq 2$ . Moments are

lim n \to \infty \frac{E (Λ_{r})}{\sqrt{n}} = \sqrt{\frac{2}{π}} G_{r, 1} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 0.498143258705129045 97... & if r = 1, 0.16722134383091813637... & if r = 2, 0.07046605176920746245... & if r = 3, 0.03218824996523203019... & if r = 4; \end{matrix}

lim n \to \infty \frac{V (Λ_{r})}{n} = G_{r, 2} - \frac{2}{π} G_{r, 1}^{2} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 0.17854905846627743895... & if r = 1, 0.02851495566901143371... & if r = 2, 0.00732819205178914862... & if r = 3, 0.00217522939296169629... & if r = 4. \end{matrix}

The mode of $Λ_{1}$ occurs at $0$ ; the median at $0.3903...$ . For $Λ_{2}$ , we did not pursue the median. The cross-correlation between $Λ_{r}$ and $N$ is

lim n \to \infty \frac{E (Λ_{r} N) - E (Λ_{r}) E (N)}{\sqrt{V (Λ_{r})} \sqrt{V (N)}} = \frac{\sqrt{1 - \frac{2}{π}} G_{r, 1}}{\sqrt{G_{r, 2} - \frac{2}{π} G_{r, 1}^{2}}} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 0.89066843... & if r = 1, 0.74816251... & if r = 2, 0.62190221... & if r = 3, 0.52141727... & if r = 4 \end{matrix}

and, again, it is possible to compute the cross-correlation between $Λ_{r}$ and $Λ_{s}$ where $r < s$ .

The mean $0.4981...$ is sharply less than the other means $0.7824...$ and $0.7978...$ we have exhibited. Why should this counterintuitive fact be true? The scenario $m / ln (n) \to 0$ is intermediate to the others. This is why we describe our work here as conjectural.

If instead $m / ln (n) \to c$ for some constant $0 < c < \infty$ , then Pavlov’s [27] density formula is

\frac{2^{c} Γ (c)}{\sqrt{2 π} Γ (2 c)} ν^{2 c} exp (\frac{- ν^{2}}{2}) .

This reduces to the density found in [16, 17] when $c = 1 / 2$ .

Given an arbitrary mapping, the deepest cycle is contained within the largest component, whereas the richest component contains the longest cycle. The deepest cycle need not be longest; the richest component need not be largest. What can be said about the probability of either event, or the average size of either structure? Questions about interplay at this level appear to be difficult to answer.

For completeness, we mention [28, 29, 30, 31, 32], which may offer additional insights and paths forward.

6 Flajolet & Odlyzko

Let $a_{n m ℓ}$ denote the number of $n$ -mappings possessing exactly $m$ components and exactly $ℓ$ cyclic points, where $n \geq ℓ \geq m \geq 2$ . We have [33, 34, 35]

\infty \sum n = ℓ n \sum ℓ = m \frac{a_{n m ℓ}}{n!} x^{n} y^{ℓ} = \frac{1}{m!} ln {(\frac{1}{1 - y τ (x)})}^{m}

where $τ (x) = x exp (τ (x))$ is Cayley’s tree function. The dominant singularity of $τ (x)$ is at $x = e^{- 1}$ and

\begin{matrix} τ (x) \sim 1 - 2^{1 / 2} \sqrt{1 - e x} & as x \to e^{- 1} . \end{matrix}

Differentiating with respect to $y$ :

\infty \sum n = ℓ n \sum ℓ = m \frac{ℓ a_{n m ℓ}}{n!} x^{n} y^{ℓ - 1} = \frac{1}{(m - 1)!} \frac{τ (x)}{1 - y τ (x)} ln {(\frac{1}{1 - y τ (x)})}^{m - 1}

and setting $y = 1$ :

	$\frac{1}{(m - 1)!} \frac{τ (x)}{1 - τ (x)} ln {(\frac{1}{1 - τ (x)})}^{m - 1}$	$\sim \frac{1}{(m - 1)!} \frac{1}{2^{1 / 2} \sqrt{1 - e x}} ln {(\frac{1}{2^{1 / 2} \sqrt{1 - e x}})}^{m - 1}$
		$\sim \frac{1}{2^{m - 1 / 2} (m - 1)!} \sqrt{\frac{1}{1 - e x}} ln {(\frac{1}{1 - e x})}^{m - 1}$

we deduce

	$n \sum ℓ = m \frac{ℓ a_{n m ℓ}}{n!}$	$\sim \frac{1}{2^{m - 1 / 2} (m - 1)!} {(\frac{1}{e})}^{- n} \frac{\sqrt{n} ln (n)^{m - 1}}{n Γ (1 / 2)}$
		$\sim \frac{ln (n)^{m - 1}}{2^{m - 1} (m - 1)!} \frac{e^{n}}{\sqrt{2 π n}}$

by the singularity analysis theorem of Flajolet & Odlyzko [36]. Multiplying both sides by $n! / n^{n}$ and using Stirling’s approximation, we obtain

n \sum ℓ = m \frac{ℓ a_{n m ℓ}}{n^{n}} \sim \frac{}{ln (n)^{m - 1}} 2^{m - 1} (m - 1)! .

From Section 4,

n \sum ℓ = m \frac{a_{n m ℓ}}{n^{n}} \sim \frac{ln (n)^{m - 1}}{2^{m - 1} (m - 1)!} \sqrt{\frac{π}{2 n}}

and hence, forming the ratio, $E (N) \sim \sqrt{2 n / π}$ as $n \to \infty$ .

Differentiating again and setting $y = 1$ :

	$\infty \sum n = ℓ n \sum ℓ = m \frac{ℓ (ℓ - 1) a_{n m ℓ}}{n!} x^{n}$	$\sim \frac{1}{(m - 1)!} {(\frac{τ (x)}{1 - τ (x)})}^{2} ln {(\frac{1}{1 - τ (x)})}^{m - 1}$
		$\sim \frac{1}{(m - 1)!} \frac{1}{2 (1 - e x)} ln {(\frac{1}{2^{1 / 2} \sqrt{1 - e x}})}^{m - 1}$
		$\sim \frac{1}{2^{m} (m - 1)!} \frac{1}{1 - e x} ln {(\frac{1}{1 - e x})}^{m - 1}$

we deduce

	$n \sum ℓ = m \frac{ℓ (ℓ - 1) a_{n m ℓ}}{n!}$	$\sim \frac{1}{2^{m} (m - 1)!} {(\frac{1}{e})}^{- n} \frac{n ln (n)^{m - 1}}{n Γ (1)}$
		$\sim \frac{ln (n)^{m - 1}}{2^{m} (m - 1)!} e^{n} .$

Multiplying by $n! / n^{n}$ and via the preceding, we obtain

n \sum ℓ = m \frac{ℓ^{2} a_{n m ℓ}}{n^{n}} \sim \frac{ln (n)^{m - 1}}{2^{m} (m - 1)!} \sqrt{2 π n} \sim \frac{ln (n)^{m - 1}}{2^{m - 1} (m - 1)!} \sqrt{\frac{π n}{2}} .

Forming the ratio, $E (N^{2}) \sim n$ as $n \to \infty$ and thus $V (N) \sim (1 - 2 / π) n$ .

7 Addendum: Divisibility

Let $m$

be a positive integer. A random variable

X

m

-divisible if it can be written as

X = Y_{1} + Y_{2} + \dots + Y_{m}

, where

Y_{1}, Y_{2}, \dots, Y_{m}

are independent and identically distributed. A random variable is infinitely divisible if it is

m

-divisible for every

m

. We wish to study the allocation of

X = N

cyclic points among a fixed number

m

of components, given a constrained random mapping. This would be a matter of determining the inverse Laplace transform of the

m^{th}

root of

L [\sqrt{\frac{2}{π}} exp (\frac{- ξ^{2}}{2})] = exp (\frac{η^{2}}{2}) erfc (\frac{η}{\sqrt{2}}) .

Pavlov’s work [24, 27] is crucial here. We confront, however, a surprising theoretical obstacle: the half-normal density is provably not infinitely divisible [37, 38, 39]. The independence requirement fails, in fact, beginning at $m = 2$ . Let us offer a plausibility argument supporting this latter assertion.

On the one hand, if $Y_{2} = 0$ is fixed, then the density of $Y_{1} = N$ clearly approaches $\sqrt{2 / π}$ as $ξ \to 0^{+}$ , i.e., it is bounded near the origin.

On the other hand, if no condition is placed on $Y_{2}$ , then the density of $Y_{1} + Y_{2} = N$ is a convolution in the $ξ$ -domain, which becomes multiplication in the $η$ -domain. Starting with [40, 41, 42]

\frac{π}{\sqrt{2 π} + π η} < \sqrt{\frac{π}{2}} exp (\frac{η^{2}}{2}) erfc (\frac{η}{\sqrt{2}}) < \frac{π}{\sqrt{2 π} + 2 η}

for all $η > 0$ , we deduce

\sqrt{\frac{1}{1 + \sqrt{\frac{π}{2}} η}} < exp (\frac{η^{2}}{4}) \sqrt{erfc (\frac{η}{\sqrt{2}})} < \sqrt{\frac{1}{1 + \sqrt{\frac{2}{π}} η}}

and upper/lower bounds are tight approximations of the center for small/large values of $η$ . No closed-form expression for $L^{- 1} [$ center $]$ seems to be possible; $L^{- 1} [$ lower bound $]$ and $L^{- 1} [$ upper bound $]$ are

\begin{matrix} \frac{2^{1 / 4}}{π^{3 / 4} \sqrt{ξ}} exp (- \sqrt{\frac{2}{π}} ξ) & and & \frac{1}{(2 π)^{1 / 4} \sqrt{ξ}} exp (- \sqrt{\frac{π}{2}} ξ) \end{matrix}

respectively. Both expressions approach infinity as $ξ \to 0^{+}$ , tentatively implying that the density of $Y_{1}$ is unbounded near the origin. This contrasts with the behavior described earlier, i.e., information about $Y_{2}$ truly affects how $Y_{1}$ is distributed. Therefore $Y_{1}$ and $Y_{2}$ must be dependent.

When we employed the word “obstacle” before, it reflected our intention to study order statistics $Z_{1} = min {Y_{1}, Y_{2}}$ and $Z_{2} = max {Y_{1}, Y_{2}}$ , with a goal of understanding the allocation process (partioning $N$ cyclic points into two cycles). If $Y_{1}$ and $Y_{2}$ were independent with common density $φ (y)$ , then the joint density of $Z_{1} \leq Z_{2}$ would simply be $2 φ (z_{1}) φ (z_{2})$ for $z_{1} \leq z_{2}$ . Dependency renders the analysis more complicated.

As an aside, the Rayleigh density is also not infinitely divisible [39]. The independence requirement again fails beginning at $m = 2$ . We argue as before, but less formally. On the one hand, if $Y_{2} = 0$ is fixed, then the density of $Y_{1}$ approaches $0$ as $ξ \to 0^{+}$ . On the other hand, if no condition is placed on $Y_{2}$ , then we wish to find the inverse Laplace transform of the square root of

L [ξ exp (\frac{- ξ^{2}}{2})] = 1 - \sqrt{\frac{π}{2}} η exp (\frac{η^{2}}{2}) erfc (\frac{η}{\sqrt{2}}) .

A remarkably accurate approximation (with error less than $0.5 %$ ):

\sqrt{1 - \sqrt{\frac{π}{2}} η exp (\frac{η^{2}}{2}) erfc (\frac{η}{\sqrt{2}})} \approx exp (\frac{η^{2}}{π}) erfc (\frac{η}{\sqrt{π}})

defies easy explanation and yet provides a very helpful estimate:

L^{- 1} [exp (\frac{η^{2}}{π}) e r f c (\frac{η}{\sqrt{π}})] = exp (\frac{- π ξ^{2}}{4})

which approaches $1$ as $ξ \to 0^{+}$ . Since $0 \neq 1$ , it follows that $Y_{1}$ and $Y_{2}$ must be dependent.

8 Addendum: Fallibility

With the benefit of hindsight, we should have focused not on $σ (x)$ , but instead on

~ σ (x) = \sqrt{x} σ (x)

both here and in [10]. The derivative of $~ σ (1 / x)$ is found as follows:

\frac{d}{d x} ~ σ (\frac{1}{x}) = \frac{d}{d x} [\frac{1}{x^{1 / 2}} σ (\frac{1}{x})] = - \frac{1}{2 x^{3 / 2}} σ (\frac{1}{x}) + \frac{1}{x^{1 / 2}} σ^{'} (\frac{1}{x}) (- \frac{1}{x^{2}})

but

σ^{'} (y) = - \frac{1}{2 y} (σ (y) + σ (y - 1))

therefore

	$\frac{d}{d x} ~ σ (\frac{1}{x})$	$= - \frac{1}{2 x^{3 / 2}} σ (\frac{1}{x}) + \frac{1}{x^{1 / 2}} (- \frac{x}{2}) [σ (\frac{1}{x}) + σ (\frac{1}{x} - 1)] (- \frac{1}{x^{2}})$
		$= - \frac{1}{2 x^{3 / 2}} σ (\frac{1}{x}) + \frac{1}{2 x^{3 / 2}} [σ (\frac{1}{x}) + σ (\frac{1}{x} - 1)] = \frac{1}{2 x^{3 / 2}} σ (\frac{1 - x}{x}),$

as was to be shown. Finally, $ρ (x)$ and $σ (x)$ are subsumed by the general DDE [4, 5, 6]

\begin{matrix} x g^{'} (x) + (1 - θ) g (x) + θ g (x - 1) = 0 % for x > 1, & g (x) = x^{θ - 1} for 0 < x \leq 1 \end{matrix}

where $θ > 0$ is fixed, and its associated Laplace transform is

\begin{matrix} L [g (ξ)] = \frac{exp (- θ E (η))}{η^{θ} / Γ (θ)}, & η \in C ∖ (- \infty, 0] . \end{matrix}

It would be good someday to learn, from an interested reader, about possible random mapping-theoretic applications of $g (x)$ for select $θ \notin {\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, \dots}$ .

9 Acknowledgements

I am grateful to Ljuben Mutafchiev [13], Lennart Bondesson [43, 44] and Jeffrey Steif [45] for helpful discussions. Vaclav Kotesovec provided relevant asymptotic expansions in [23]. The Mathematica routines NDSolve for DDEs [46] and NInverseLaplaceTransform [47] assisted in numerically confirming many results. Interest in this subject has, for me, spanned many years [48, 49].

References

[1] K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fysik, v. 22A (1930) n. 10, 1–14.
[2] P. Moree, Nicolaas Govert de Bruijn, the enchanter of friable integers, Indag. Math. 24 (2013) 774–801; arXiv:1212.1579; MR3124806.
[3] J. C. Lagarias, Euler’s constant: Euler’s work and modern developments, Bull. Amer. Math. Soc. 50 (2013) 527–628; arXiv:1303.1856; MR3090422.
[4] G. A. Watterson, The stationary distribution of the infinitely-many neutral alleles diffusion model, J. Appl. Probab. 13 (1976) 639–651; 14 (1977) 897; MR0504014 and MR0504015.
[5] R. Arratia, A. D. Barbour and S. Tavaré, Random combinatorial structures and prime factorizations, Notices Amer. Math. Soc. 44 (1997) 903–910; MR1467654.
[6] R. Arratia, A. D. Barbour and S. Tavaré, Logarithmic Combinatorial Structures: a Probabilistic, Approach, Europ. Math. Society, 2003, pp. 21-24, 52, 87–89, 118; MR2032426.
[7] S. R. Finch, Permute, Graph, Map, Derange, arXiv:2111.05720.
[8] S. R. Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487.
[9] S. R. Finch, Second best, Third worst, Fourth in line, arXiv:2202.07621.
[10] S. R. Finch, Joint probabilities within random permutations, arXiv:2203.10826.
[11] V. E. Stepanov, Limit distributions of certain characteristics of random mappings (in Russian), Teor. Verojatnost. i Primenen. 14 (1969) 639–653; Engl. transl. in Theory Probab. Appl. 14 (1969) 612–626; MR0278350.
[12] F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer-Verlag, 1973, pp. 172, 364, 367; MR0352889.
[13] L. R. Mutafchiev, Large components and cycles in a random mapping pattern, Random Graphs ’87, Proc. 1987 Poznań conf., ed. M. Karoński, J. Jaworski and A. Ruciński, Wiley, 1990, pp. 189–202; MR1094133.
[14] V. F. Kolchin, Random Mappings, Optimization Software, 1986, pp. 35, 46–48, 85–88, 93–94, 153; MR0865130.
[15] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1965, pp. 174–179; MR0176501.
[16]
H. Rubin and R. Sitgreaves, Probability distributions related to random transformations of a finite set, Stanford Univ. tech. report, 1954,
http://statistics.stanford.edu/technical-reports/probability-distributions-related-random-transformations-finite-set.
[17] B. Harris, Probability distributions related to random mappings, Annals Math. Statist. 31 (1960) 1045–1062; MR0119227.
[18] P. W. Purdom and J. H. Williams, Cycle length in a random function, Trans. Amer. Math. Soc. 133 (1968) 547–551; MR0228032.
[19] L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966) 340–357; MR0195117.
[20] R. C. Griffiths, On the distribution of allele frequencies in a diffusion model, Theoret. Population Biol. 15 (1979) 140–158; MR0528914.
[21] T. Shi, Cycle lengths of $θ$ -biased random permutations, B.S. thesis, Harvey Mudd College, 2014, http://scholarship.claremont.edu/hmc_theses/65/.
[22] A. Rényi, On connected graphs. I, Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 (1959) 385–388; MR0126842.
[23] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A001865, A060281, A065456, and A273434.
[24] Yu. L. Pavlov, Limit theorems for a characteristic of a random mapping (in Russian), Teor. Veroyatnost. i Primenen. 26 (1981) 841–847; Engl. transl. in Theory Probab. Appl. 26 (1982) 829–834; MR0636781.
[25] M. D. Kruskal, The expected number of components under a random mapping function, Amer. Math. Monthly 61 (1954) 392–397; MR0062973.
[26] J. Riordan, Enumeration of linear graphs for mappings of finite sets, Annals Math. Statist. 33 (1962) 178–185; MR0133250.
[27] Yu. L. Pavlov, Random mappings with constraints on the number of cycles (in Russian), Trudy Mat. Inst. Steklov. 177 (1986) 122–132, 208; Engl. transl. in Proc. Steklov Inst. Math. 1988, n. 4, 131–142; MR0840680.
[28] J. M. DeLaurentis, Random functions and their small cycles, Proc. 18 $^{th}$ Southeastern Conf. on Combinatorics, Graph Theory and Computing, Boca Raton, 1987, ed. F. Hoffman, R. C. Mullin, R. G. Stanton and K. Brooks Reid, Congr. Numer. 58, Utilitas Math., 1987, pp. 295–302; MR0944710.
[29] J. M. DeLaurentis, Components and cycles of a random function, Advances in Cryptology – CRYPTO ’87, Proc. 1987 Santa Barbara conf., Lect. Notes in Comp. Sci. 293, Springer-Verlag, 1988, pp. 231–242; MR0956654.
[30] D. Romero and F. Zertuche, The asymptotic number of attractors in the random map model, J. Phys. A 36 (2003) 3691–3700; MR1984723.
[31] D. Romero and F. Zertuche, Grasping the connectivity of random functional graphs, Studia Sci. Math. Hungar. 42 (2005) 1–19; MR2128671.
[32] R. S. V. Martins, D. Panario, C. Qureshi and E. Schmutz, Periods of iterations of functions with restricted preimage sizes, ACM Trans. Algorithms 16 (2020) A. 30; arXiv:1701.09148; MR4120518.
[33] J. Kupka, The distribution and moments of the number of components of a random function, J. Appl. Probab. 27 (1990) 202–207; MR1039196.
[34] B. J. English, Factorial moments for random mappings by means of indicator variables, J. Appl. Probab. 30 (1993) 167–174; MR1206359.
[35] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A201685.
[36] P. Flajolet and A. M. Odlyzko, Random mapping statistics, Advances in Cryptology - EUROCRYPT ’89, ed. J.-J. Quisquater and J. Vandewalle, Lect. Notes in Comp. Sci. 434, Springer-Verlag, 1990, pp. 329–354; MR1083961.
[37] A. Ruegg, A characterization of certain infinitely divisible laws, Annals Math. Statist. 41 (1970) 1354–1356; MR0267626.
[38] E. Lukacs,
Developments in Characteristic Function Theory
, Macmillan Co., 1983, pp. 68–70; MR0810001.
[39] F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Dekker, 2004, pp. 77, 113, 125–126, 411–412; MR.2011862.
[40] Z. W. Birnbaum, An inequality for Mill’s ratio, Annals Math. Statist. 13 (1942) 245–246; MR0006640.
[41] Y. Komatu, Elementary inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. 4 (1955) 69–70; MR0079844.
[42] A. Gasull and F. Utzet, Approximating Mills ratio, J. Math. Anal. Appl. 420 (2014) 1832–1853; MR3240110.
[43] L. Bondesson, On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line, Scand. Actuar. J. (1987) 225–247; MR0943583.
[44] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lect. Notes in Stat. 76, Springer-Verlag, 1992, p. 67; MR1224674.
[45] B. G. Franzén, J. E. Steif and J. Wästlund, Where to stand when playing darts? ALEA Latin Amer. J. Probab. Math. Statist. 18 (2021) 1561–1583; arXiv:2010.00566; MR4291033.
[46] Y. M. Beltukov, Tuning NDSolve options to accurately solve DDEs, http://mathematica.stackexchange.com/questions/59431/simpler-code-evaluating-dickmans-function.
[47] P. Valko, NInverseLaplaceTransform numerical approximation package, http://resources.wolframcloud.com/FunctionRepository/resources/NInverseLaplaceTransform/.
[48] S. R. Finch, Golomb-Dickman constant, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 284–292; MR2003519.
[49] S. R. Finch, Extreme prime factors, Mathematical Constants II, Cambridge Univ. Press, 2019, pp. 171–172; MR3887550.

Steven Finch

MIT Sloan School of Management

Cambridge, MA, USA

steven_finch@harvard.edu

Components and Cycles of Random Mappings

On non-Hamiltonian cycle sets of satisfying Grinberg's Equation

Short cycle covers of cubic graphs and intersecting 5-circuits

Permute, Graph, Map, Derange

Second Best, Third Worst, Fourth in Line

Longest Cycle above Erdős-Gallai Bound

Period-halving Bifurcation of a Neuronal Recurrence Equation

On inverse product cannibalisation: a new Lotka-Volterra model for asymmetric competition in the ICTs

1 Stepanov

2 Mutafchiev

3 Purdom & Williams

4 Rényi

5 Pavlov

6 Flajolet & Odlyzko

7 Addendum: Divisibility

8 Addendum: Fallibility

9 Acknowledgements

References

Components and Cycles of Random Mappings

Related Research

On non-Hamiltonian cycle sets of satisfying Grinberg's Equation

Short cycle covers of cubic graphs and intersecting 5-circuits

Permute, Graph, Map, Derange

Second Best, Third Worst, Fourth in Line

Longest Cycle above Erdős-Gallai Bound

Period-halving Bifurcation of a Neuronal Recurrence Equation

On inverse product cannibalisation: a new Lotka-Volterra model for asymmetric competition in the ICTs

1 Stepanov

2 Mutafchiev

3 Purdom & Williams

4 Rényi

5 Pavlov

6 Flajolet & Odlyzko

7 Addendum: Divisibility

8 Addendum: Fallibility

9 Acknowledgements

References