Joint Probabilities within Random Permutations

1 Density

Difficulties presented by the numerical integration of $f_{12} (x, y)$ are evident in Figure 2. The surface appears to touch the $x y$ -plane only when $y = 0$ ; its prominent ridge occurs along the line $y = (1 - x) / 2$ because $(1 - x - y) / y = 1$ corresponds to a unique point of nondifferentiability for $ξ \mapsto ρ (ξ)$ ; its remaining boundary hovers over the broken line $y = min {x, 1 - x}$ , everywhere finite except in the vicinity of $x = 0$ .

Figure 2: Probability density of $(Λ_{1}, Λ_{2})$ , over $0 \leq y \leq 1 / 2$ and $y \leq x \leq 1 - y .$

Complications are compounded for the three other densities (which are, in themselves, approximations). Figure 3 contains a plot of

f_{13} (x, z) = x \int z f_{123} (x, y, z) d y .

The surface appears to touch the $x z$ -plane when $z = 0$ and $0 < x < 1 / 2$ simultaneously, as well as everywhere along the broken line $z = min {x, (1 - x) / 2}$ .

Figure 3: Probability density of $(Λ_{1}, Λ_{3})$ , over $0 \leq z \leq 1 / 3$ and $z \leq x \leq 1 - 2 z .$

Figure 4 contains a plot of

f_{14} (x, w) = min {x, 1 / 3} \int w x \int z f_{1234} (x, y, z, w) d y d z .

The (precipitously rising) surface appears to touch the $x w$ -plane only when $w = 0$ and $0 < x < 1 / 2$ simultaneously; its remaining boundary hovers over the broken line $w = min {x, (1 - x) / 3}$ , everywhere finite except in the vicinity of $x = 0$ . The vertical scale is more expansive here than for the other plots.

Figure 4: Probability density of $(Λ_{1}, Λ_{4})$ , over $0 \leq w \leq 1 / 4$ and $w \leq x \leq 1 - 3 w .$

Figure 5 contains a plot of

f_{23} (y, z) = 1 \int y f_{123} (x, y, z) d x .

The (fairly undulating) surface appears to touch the $y z$ -plane only when $z = 1 - 2 y$ . Unlike the other densities, a singularity here occurs at $(y, z) = (0, 0)$ .

Figure 5: Probability density of $(Λ_{2}, Λ_{3})$ , over $0 \leq z \leq 1 / 3$ and $z \leq y \leq (1 - z) / 2.$

2 Correlation

Let

\begin{matrix} E (x) = \infty \int x \frac{e^{- t}}{t} d t = - Ei (- x), & x > 0 \end{matrix}

be the exponential integral. Upon normalization, the $h^{th}$ moment of the $r^{th}$ longest cycle length is [10, 11, 12]

lim n \to \infty \frac{E (Λ_{r}^{h})}{n^{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 $2$ ). The cross-correlation between $r^{th}$ longest and $s^{th}$ longest cycle lengths is

	$κ_{r, s}$	$= \frac{E (Λ_{r} Λ_{s}) - E (Λ_{r}) E (Λ_{s})}{\sqrt{E (Λ_{r}^{2}) - E {(Λ_{r})}_{r}^{2}} \sqrt{E (Λ_{s}^{2}) - E {(Λ_{s})}_{s}^{2}}}$
		$\to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - 0.75803584... & if r = 1 and s = 2, - 0.78421290... & if r = 1 and s = 3, - 0.68442819... & if r = 1 and s = 4, + 0.35549741... & if r = 2 and s = 3 \end{matrix}$

with cross-moments given by [13, 14]

lim n \to \infty \frac{E (Λ_{1} Λ_{2})}{n^{2}} = \frac{1}{2} \infty \int 0 x \int 0 exp [- E (y) - x - y] d y d x,

lim n \to \infty \frac{E (Λ_{1} Λ_{3})}{n^{2}} = \frac{1}{2} \infty \int 0 x \int 0 y \int 0 \frac{1}{y} exp [- E (z) - x - y - z] d z d y d x,

lim n \to \infty \frac{E (Λ_{1} Λ_{4})}{n^{2}} = \frac{1}{2} \infty \int 0 x \int 0 y \int 0 z \int 0 \frac{1}{y z} exp [- E (w) - x - y - z - w] d w d z d y d x,

lim n \to \infty \frac{E (Λ_{2} Λ_{3})}{n^{2}} = \frac{1}{2} \infty \int 0 x \int 0 y \int 0 \frac{1}{x} exp [- E (z) - x - y - z] d z d y d x .

The fact that $Λ_{1}$ is negatively correlated with other $Λ_{r}$ , yet $Λ_{2}$ is positively correlated with other $Λ_{s}$ , is due to longest cycles typically occupying a giant-size portion of permutations, but second-longest cycles less so.

3 Distribution

Bach & Peralta [15]

discussed a remarkable heuristic model, based on random bisection, that simplifies the computation of joint probabilities involving

Λ_{1}

and

Λ_{2}

. In the same paper, they rigorously proved that asymptotic predictions emanating from the model are valid. Subsequent researchers extended the work to

Λ_{1}

and

Λ_{3}

, to

Λ_{1}

and

Λ_{4}

, and to

Λ_{2}

and

Λ_{3}

. We shall not enter into details of the model nor its absolute confirmation, preferring instead to dwell on numerical results and certain relative verifications.

3.1 First and Second

For $0 < a \leq b \leq 1$ , Bach & Peralta [15] demonstrated that

limn→∞P{Λ2n≤a \& Λ1n≤b}=ρ(1a)=I0(a)+b∫aρ(1−xa)dxx=I1(a,b).

Note the slight change from earlier – writing $Λ_{2}$ before $Λ_{1}$ – a convention we adopt so as to be consistent with the literature. Let $J_{1} (a, b) = I_{0} (a) + I_{1} (a, b)$ . Return now to the example from the introduction. Evaluating

J_{1} (\frac{1}{3}, \frac{1}{2}) = ρ (3) + 1 / 2 \int 1 / 3 ρ (\frac{1 - x}{1 / 3}) \frac{d x}{x}

is less numerically problematic than evaluating

1 / 3 \int 0 x \int 0 f_{12} (x, y) d y d x      = ρ (3) + 1 / 2 \int 1 / 3 1 / 3 \int 0 f_{12} (x, y) d y d x

for two reasons:

a double integral has been miraculously reduced to a single integral,
the argument of $ρ$ within the integral is $(1 - x) / a$ rather than $(1 - x - y) / y$ , which is unstable as $y \to 0$ .

The advantages of using the Bach & Peralta formulation will become more apparent as we move forward (incidently, their $G$ is the same as our $J_{1}$ ).

$u ∖ v$	$1$	$1$	$2$	$3$	$4$	$5$
$2$	0.30685282	0.69314718
$3$	0.04860839	0.80417093	0.17604345
$4$	0.00491093	0.61877013	0.09148808	0.01974468
$5$	0.00035472	0.46286746	0.03043740	0.00578984	0.00149456
$6$	0.00001965	0.36519810	0.00849154	0.00107262	0.00029307	0.00008552

Table 1: $I_{0} (1 / u)$ and $I_{1} (1 / u, 1 / v)$ for $2 \leq u \leq 6$ , $1 \leq v < u$

$u ∖ v$	$1$	$2$	$3$	$4$	$5$	$6$
$2$	1.00000000	0.30685282
$3$	0.85277932	0.22465184	0.04860839
$4$	0.62368106	0.09639901	0.02465561	0.00491093
$5$	0.46322219	0.03079212	0.00614457	0.00184928	0.00035472
$6$	0.36521775	0.00851119	0.00109227	0.00031272	0.00010517	0.00001965

Table 2: $J_{1} (1 / u, 1 / v)$ for $2 \leq u \leq 6$ , $1 \leq v \leq u$

A verification of $J_{1} (a, b)$ is as follows:

\frac{\partial J_{1}}{\partial b} = ρ (\frac{1 - b}{a}) \frac{1}{b}

by the Second Fundamental Theorem of Calculus, hence

as anticipated by Billingsley [5]. An interpretation of $I_{1} (a, b)$ is helpful:

I1(a,b)=limn→∞P{Λ2n≤a \& a<Λ1n≤b}

i.e., the probability that exactly one cycle has length in the interval $(a n,$ $b n]$ and all others have length $\leq a n$ . We have, for instance,

\begin{matrix} {\frac{\partial I_{1}}{\partial a} ∣ ∣ ∣}_{b = 1} = 0, & I_{1} (a, 1) \approx 0.8285 \end{matrix}

when $a \approx 0.3775 \approx 1 / (2.649)$ , the value maximizing $P {Λ_{2} \leq a n < Λ_{1}}$ as $n \to \infty$ .

3.2 First and Third

For $0 < a \leq 1 / 2$ and $a \leq b \leq 1$ , Lambert [16] demonstrated that

J2(a,b)=limn→∞P{Λ3n≤a \& Λ1n≤b}=J1(a,b)+b∫ab∫yρ(1−x−ya)dxxdyy=I2(a,b).

(Incidently, his $G_{2}$ is the same as our $J_{2} - J_{1} = I_{2}$ .)

$u ∖ v$	$1$	$2$	$3$	$4$	$5$
$3$	0.14722068	0.08220098
$4$	0.36143259	0.19556747	0.01998464
$5$	0.46463747	0.20709082	0.02278925	0.00201596
$6$	0.48588944	0.16644726	0.01263312	0.00136571	0.00013356

Table 3: $I_{2} (1 / u, 1 / v)$ for $3 \leq u \leq 6$ , $1 \leq v < u$

$u ∖ v$	$1$	$2$	$3$	$4$	$5$	$6$
$3$	1.00000000	0.30685282	0.04860839
$4$	0.98511365	0.29196647	0.04464025	0.00491093
$5$	0.92785965	0.23788294	0.02893382	0.00386524	0.00035472
$6$	0.85110720	0.17495845	0.01372538	0.00167843	0.00023872	0.00001965

Table 4: $J_{2} (1 / u, 1 / v)$ for $3 \leq u \leq 6$ , $1 \leq v \leq u$

A verification of $J_{2} (a, b)$ is as follows:

	$\frac{\partial I_{2}}{\partial b}$	$= \frac{1}{2} \frac{\partial}{\partial b} b \int a b \int a ρ (\frac{1 - x - y}{a}) \frac{d x}{x} \frac{d y}{y}$
		$= \frac{1}{2} b \int a ρ (\frac{1 - b - y}{a}) \frac{1}{b} \frac{d y}{y} + \frac{1}{2} b \int a ρ (\frac{1 - x - b}{a}) \frac{1}{b} \frac{d x}{x} = b \int a ρ (\frac{1 - x - b}{a}) \frac{1}{b} \frac{d x}{x}$

by symmetry; thus by Leibniz’s Rule,

	$\frac{\partial^{2} I_{2}}{\partial a \partial b}$	$= - b \int a ρ^{'} (\frac{1 - x - b}{a}) \frac{1 - x - b}{a^{2}} \frac{1}{b} \frac{d x}{x} - ρ (\frac{1 - a - b}{a}) \frac{1}{a b}$
		$= b \int a \frac{ρ (\frac{1 - a - x - b}{a})}{\frac{1 - x - b}{a}} \frac{1 - x - b}{a^{2} x b} d x - \frac{\partial^{2} J_{1}}{\partial a \partial b}$

hence

\frac{\partial^{2} J_{2}}{\partial a \partial b} = b \int a \frac{ρ (\frac{1 - a - x - b}{a})}{a x b} d x = b \int a f_{123} (b, x, a) d x = f_{13} (b, a),

as was to be shown. An interpretation of $I_{2} (a, b)$ is helpful:

I2(a,b)=limn→∞P{Λ3n≤a \& a<Λ2n≤Λ1n≤b}

i.e., the probability that exactly two cycles have length in the interval $(a n,$ $b n]$ and all others have length $\leq a n$ .

3.3 First and Fourth

For $0 < a \leq 1 / 3$ and $a \leq b \leq 1$ , Cavallar [17] and Zhang [18] independently demonstrated that

J3(a,b)=limn→∞P{Λ4n≤a \& Λ1n≤b}=J2(a,b)+b∫ab∫zb∫yρ(1−x−y−za)dxxdyydzz=I3(a,b).

(Incidently, Cavallar’s $G_{3}$ is the same as our $J_{3} - J_{2} = I_{3}$ while Zhang’s $G_{3}$ is the same as our $J_{3}$ .)

$u ∖ v$	$1$	$2$	$3$	$4$	$5$
$4$	0.01488635	0.01488635	0.00396814
$5$	0.07126587	0.06809540	0.01884107	0.00094238
$6$	0.14082221	0.12382378	0.02870816	0.00222512	0.00009015

Table 5: $I_{3} (1 / u, 1 / v)$ for $4 \leq u \leq 6$ , $1 \leq v < u$

$u ∖ v$	$1$	$2$	$3$	$4$	$5$	$6$
$4$	1.00000000	0.30685282	0.04860839	0.00491093
$5$	0.99912552	0.30597834	0.04777489	0.00480762	0.00035472
$6$	0.99192941	0.29878222	0.04243355	0.00390355	0.00032887	0.00001965

Table 6: $J_{3} (1 / u, 1 / v)$ for $4 \leq u \leq 6$ , $1 \leq v \leq u$

We omit details of the verification of $J_{3} (a, b)$ , except to mention the start point

\frac{\partial I_{3}}{\partial b} = \frac{1}{6} \frac{\partial}{\partial b} b \int a b \int a b \int a ρ (\frac{1 - x - y - z}{a}) \frac{d x}{x} \frac{d y}{y} \frac{d z}{z}

and the end point $\partial^{2} J_{3} / \partial a \partial b = f_{14} (b, a)$ . An interpretation of $I_{3} (a, b)$ is helpful:

I3(a,b)=limn→∞P{Λ4n≤a \& a<Λ3n≤Λ1n≤b}

i.e., the probability that exactly three cycles have length in the interval $(a n,$ $b n]$ and all others have length $\leq a n$ .

3.4 Second and Third

For $0 < a < 1 / 3$ , $a \leq b < 1 / 2$ and $b \leq c \leq 1$ , Ekkelkamp [19, 20] demonstrated that

limn→∞P{Λ3n≤a, a<Λ2n≤b \& Λ1n≤c}=b∫ac∫yρ(1−x−ya)dxxdyy

under the additional condition $a + b + c \leq 1$ . If we were to suppose that this condition is unnecessary and set $c = 1$ , then by definition of $ρ_{2}$ , we would have

L1(a,b)=limn→∞P{Λ3n≤a \& Λ2n≤b}=ρ2(1a)=K0(a)+b∫a1∫yρ1(1−x−ya)dxxdyy=K1(a,b)

where $K_{1}$ is similar (but not identical) to $I_{2}$ :

K1(a,b)=limn→∞P{Λ3n≤a \& a<Λ2n≤b}.

On the one hand, our supposition is evidently false. In the following, we compare provisional theoretical values (eight digits of precision) against simulated values (just two digits):

$u ∖ v$	$3$	$3$	$4$	$5$
$4$	0.62368106	0.27362816 $>$ 0.21
$5$	0.46322219	0.40043992 $>$ 0.32	0.17285583 $>$ 0.14
$6$	0.36521775	0.43489680 $>$ 0.35	0.24479052 $>$ 0.20	0.10650591 $>$ 0.09

Table 7: $K_{0} (1 / u)$ and $K_{1} (1 / u, 1 / v)$ for $4 \leq u \leq 6$ , $3 \leq v < u$

$u ∖ v$	$2$	$3$	$4$	$5$	$6$
$3$	1.00000000	0.85277932
$4$	0.98511365	0.89730922 $>$ 0.84	0.62368106
$5$	0.92785965	0.86366210 $>$ 0.79	0.63607802 $>$ 0.60	0.46322219
$6$	0.85110720	0.80011455 $>$ 0.72	0.61000827 $>$ 0.56	0.47172366 $>$ 0.45	0.36521775

Table 8: $L_{1} (1 / u, 1 / v)$ for $3 \leq u \leq 6$ , $2 \leq v \leq u$

where special cases

L_{1} (a, b) = {\begin{matrix} ρ_{2} (1 / b) & if a = b \leq 1 / 3, ρ_{3} (1 / a) & if a \leq 1 / 3 and b = 1 / 2 \end{matrix}

are surely true.

On the other hand, a verification of $L_{1} (a, b)$ is as follows:

\frac{\partial L_{1}}{\partial b} = \frac{\partial}{\partial b} b \int a 1 \int y ρ (\frac{1 - x - y}{a}) \frac{d x}{x} \frac{d y}{y} = 1 \int b ρ (\frac{1 - x - b}{a}) \frac{1}{b} \frac{d x}{x}

hence by Leibniz’s Rule,

	$\frac{\partial^{2} L_{1}}{\partial a \partial b}$	$= - 1 \int b ρ^{'} (\frac{1 - x - b}{a}) \frac{1 - x - b}{a^{2}} \frac{1}{b} \frac{d x}{x} = 1 \int b \frac{ρ (\frac{1 - a - b - x}{a})}{\frac{1 - b - x}{a}} \frac{1 - b - x}{a^{2} b x} d x$
		$= 1 \int b \frac{ρ (\frac{1 - a - b - x}{a})}{a b x} d x = 1 \int b f_{123} (x, b, a) d x = f_{23} (b, a),$

as was to be shown. If a correction term of the form $φ (a) + ψ (b)$ could be incorporated into $K_{1} (a, b)$ , rendering it suitably smaller, then the above argument would still go through. Determining such expressions $φ (a)$ , $ψ (b)$ is an open problem.

For $0 < α < 1 / 4$ , $α \leq β < 1 / 3$ , $β \leq γ < 1 / 2$ and $γ \leq δ \leq 1$ , Ekkelkamp [19, 20] further demonstrated that

	$limn→∞P{Λ4n≤α, α<Λ3n≤β, β<Λ2n≤γ \& Λ1n≤δ}$
	$= β \int α γ \int z δ \int y ρ (\frac{1 - x - y - z}{α}) \frac{d x}{x} \frac{d y}{y} \frac{d z}{x}$

under the additional condition $α + β + γ + δ \leq 1$ . Such a formula might eventually assist in calculating

limn→∞P{Λ4n≤α \& Λ2n≤γ},limn→∞P{Λ4n≤α \& Λ3n≤β}.

We leave this task for others. Accuracy can be improved by including a subordinate term – we have studied only main terms of asymptotic expansions – this fact was mentioned in [21], citing [19], but for proofs one must refer to [20]. It is striking that so much of this material remains unpublished (seemingly abandoned but thankfully preserved in doctoral dissertations; see [22, 23] for more).

An odd confession is necessary at this point and it is almost surely overdue. The multivariate probabilities discussed here were originally conceived not in the context of

n

-permutations as

n \to \infty

, but instead in the difficult realm of integers

\leq N

(prime factorizations with cryptographic applications) as

N \to \infty

. Knuth & Trabb Pardo [3, 24, 25] were the first to tenuously observe this analogy. Lloyd [26, 27] reflected, “They do not explain the coincidence… No isomorphism of the problems is established”. Early in his article, Tao [28] wrote how a certain calculation doesn’t offer understanding for “why there is such a link”, but later gave what he called a “satisfying conceptual (as opposed to computational) explanation”. After decades of waiting, the fog has apparently lifted.

4 Addendum: Mappings

A counterpart of Billingsley’s $f_{1234}$ :

g_{1234} (x, y, z, w) = \frac{1}{16 x y z w} σ (\frac{1 - x - y - z - w}{w}) \frac{1}{\sqrt{w}},

\begin{matrix} 1 > x > y > z > w > 0, & x + y + z + w < 1; \end{matrix}

\begin{matrix} ξ σ^{'} (ξ) + \frac{1}{2} σ (ξ) + \frac{1}{} 2 σ (ξ - 1) = 0 for ξ > 1, & σ (ξ) = 1 / \sqrt{ξ} for 0 < ξ \leq 1 \end{matrix}

is applicable to the study of connected components in random mappings [6, 8]. Let $Λ_{1}$ and $Λ_{2}$ denote the largest and second-largest such components. We use similar notation, but different techniques (because not as much is known about $σ$ as about $ρ$ .) For example,

	$lim n \to \infty P {\frac{Λ_{1}}{n} > \frac{1}{2}}$	$= 1 \int 1 / 2 g_{1} (x) d x = 1 \int 1 / 2 \frac{1}{2 x} σ (\frac{1 - x}{x}) \frac{d x}{\sqrt{x}}$
		$= \frac{1}{2} 1 \int 1 / 2 \frac{1}{x \sqrt{1 - x}} d x = ln (1 + \sqrt{2}) .$

Call this probability $Q$ . The analog here of what we called $A$ in the introduction is

	$1−limn→∞P{Λ1n>12}−limn→∞P{Λ1n≤12 \& 13<Λ2n≤12}$
	$= 1 - Q - 1 / 2 \int 1 / 3 x \int 1 / 3 g_{12} (x, y) d y d x = 1 - Q - 1 / 2 \int 1 / 3 x \int 1 / 3 \frac{1}{4 x y} σ (\frac{1 - x - y}{y}) \frac{d y d x}{\sqrt{y}}$
	$= 1 - Q - \frac{1}{4} 1 / 2 \int 1 / 3 x \int 1 / 3 \frac{d y d x}{x y \sqrt{1 - x - y}} = 0.065484671 719...$

and the analog of we called $1 - A - B$ is

	$limn→∞P{Λ1n>12}−limn→∞P{Λ1n>12 \& 13<Λ2n≤12}$
	$= Q - 2 / 3 \int 1 / 2 1 - x \int 1 / 3 g_{12} (x, y) d y d x = Q - 2 / 3 \int 1 / 2 1 - x \int 1 / 3 \frac{1}{4 x y} σ (\frac{1 - x - y}{y}) \frac{d y d x}{\sqrt{y}}$
	$= Q - \frac{1}{4} 2 / 3 \int 1 / 2 1 - x \int 1 / 3 \frac{d y d x}{x y \sqrt{1 - x - y}} = 0.7800879 54710....$

Thus the analog of $B$ (associated with the orange $\cup$ brown triangle in Figure 1) is

lim n \to \infty P {\frac{Λ_{2}}{n} > \frac{1}{3}} = 1 - A - (1 - A - B) = 0.154427373569...

and should lead in due course to a formula for $σ_{2}$ , generalizing $σ_{1} = σ$ .

Figure 6: $f_{1} (x) = \frac{1}{x} ρ (\frac{1 - x}{x})$ and $g_{1} (x) = \frac{1}{2 x^{3 / 2}} σ (\frac{1 - x}{x})$ comparison;the differential expression $g_{1} (x) = \frac{d}{d x} (\frac{1}{x^{1 / 2}} σ (\frac{1}{x}))$ is akin to $f_{1} (x) = \frac{d}{d x} ρ (\frac{1}{x})$ .

Figure 7: $g_{12} (x, y) = \frac{1}{4 x y^{3 / 2}} σ (\frac{1 - x - y}{y})$ over $0 \leq y \leq 1 / 2$ and $y \leq x \leq 1 - y$ ; this contrasts sharply from plot of $f_{12} (x, y)$ in Figure 2 along diagonal segment $x = y$ .

5 Addendum: Short Cycles

Given a random $n$ -permutation, let $S_{r}$ denote the length of the $r^{th}$ shortest cycle ( $0$ if the permutation has no $r^{th}$ cycle) and $C_{ℓ}$ denote the number of cycles of length $ℓ$ . Since, as $n \to \infty$ , the distribution of $C_{ℓ}$ approaches Poisson( $1 / ℓ$ ) and $C_{1}$ , $C_{2}$ , $C_{3}$ , … become asymptotically independent [29], we can calculate corresponding probabilities for $S_{r}$ . For example,

P {S_{1} = 1} = P {C_{1} \geq 1} = 1 - P {C_{1} = 0} = 1 - e^{- 1},

	$P {S_{1} = 2}$	$=P{C1=0 \&\ C2≥1}=P{C1=0}−P{C1=0 \&\ C2=0}$
		$= P {C_{1} = 0} (1 - P {C_{2} = 0}) = e^{- 1} (1 - e^{- 1 / 2}) = e^{- 1} - e^{- 3 / 2}$

and, more generally,

\begin{matrix} P {S_{1} = i} = e^{- H_{i - 1}} - e^{- H_{i}}, & H_{m} = m \sum k = 1 \frac{1}{k} . \end{matrix}

It is understood that these are limiting quantities as $n \to \infty$ . As another example,

P {S_{2} = 1} = P {C_{1} \geq 2} = 1 - P {C_{1} \leq 1} = 1 - 2 e^{- 1},

	$P {S_{2} = 2}$	$=P{C1=1 \&\ C2≥1}+P{C1=0 \&\ C2≥2}$
		$=P{C1=1}−P{C1=1 \&\ C2=0}+P{C1=0}−P{C1=0 \&\ C2≤1}$
		$= e^{- 1} (1 - e^{- 1 / 2}) + e^{- 1} (1 - \frac{3}{2} e^{- 1 / 2}) = 2 e^{- 1} - \frac{5}{2} e^{- 3 / 2}$

and

P {S_{2} = j} = (H_{j - 1} + 1) e^{- H_{j - 1}} - (H_{j} + 1) e^{- H_{j}} .

Similar reasoning leads to

P{S1=i \&\ S2=j}=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩e−Hi−1−(1+1i)e−Hiif i=j,1i(e−Hj−1−e−Hj)if i<j,0otherwise

enabling a conjecture: $E (S_{1} S_{2}) = O (ln (n)^{3})$ . A proof still remains out of reach.

6 Acknowledgements

I am grateful to Michael Rogers, Josef Meixner, Nicholas Pippenger, Eran Tromer, John Kingman, Andrew Barbour, Ross Maller and Joseph Blitzstein for helpful discussions. The creators of Mathematica, as well as administrators of the MIT Engaging Cluster, earn my gratitude every day. Interest in this subject has, for me, spanned many years [30, 31]. A sequel to this paper will be released soon [32].

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Steven Finch

MIT Sloan School of Management

Cambridge, MA, USA

steven_finch@harvard.edu

Joint Probabilities within Random Permutations

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Taxonomy and Practical Evaluation of Primality Testing Algorithms

Survey schemes for stochastic gradient descent with applications to M-estimation

Asymptotic Equivalence of Fixed-size and Varying-size Determinantal Point Processes

1 Density

2 Correlation

3 Distribution

3.1 First and Second

3.2 First and Third

3.3 First and Fourth

3.4 Second and Third

4 Addendum: Mappings

5 Addendum: Short Cycles

6 Acknowledgements

References

Joint Probabilities within Random Permutations

Related Research

Random primes in arithmetic progressions

Probabilities of first order sentences on sparse random relational structures: An application to definability on random CNF formulas

Stability of Sampling for CUR Decompositions

Taxonomy and Practical Evaluation of Primality Testing Algorithms

Survey schemes for stochastic gradient descent with applications to M-estimation

Asymptotic Equivalence of Fixed-size and Varying-size Determinantal Point Processes

1 Density

2 Correlation

3 Distribution

3.1 First and Second

3.2 First and Third

3.3 First and Fourth

3.4 Second and Third

4 Addendum: Mappings

5 Addendum: Short Cycles

6 Acknowledgements

References