DeepAI

# Components and Cycles of Random Mappings

Each connected component of a mapping {1,2,...,n}→{1,2,...,n} contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint – that exactly one component exists – are also examined. For instance, the mean length of the longest cycle is (0.7824...)√(n) in general, but for the special case, it is (0.7978...)√(n), a difference of less than 2%.

06/24/2019

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## 1 Stepanov

For introductory purposes, let us examine solely one-to-one mappings , i.e., permutations on symbols.  Let denote the length of the longest cycle in an -permutation, chosen uniformly at random.  We have

 limn→∞P{Λ≤an}=ρ(1a),0

where, from the DDE,

 ρ(x)=⎧⎪ ⎪⎨⎪ ⎪⎩1−ln(x)if 1

Thus, for example,

 limn→∞P{13<Λn≤12} =ρ(2)−ρ(3) =π212−ln(2)+ln(3)−12ln(3)2−Li2(13) =0.258244431148....

Let us now explore a less-familiar approach [11].  Define a function to be equal to for and for .  Using the series expansion for in terms of , we have

 L−1[exp(−E(η))η]=∞∑k=0(−1)kk!L−1[E(η)kη]

where the power is the Laplace transform of the -fold self-convolution of .  On the one hand, formulas

are well-known.  On the other hand,

 L−1[E(η)2η]=⎧⎪ ⎪⎨⎪ ⎪⎩lll−π26+ln(ξ)2+2Li2(1ξ)if ξ≥2,0if 0≤ξ<2.

does not appear in [12].  Since for , only terms of the series need to be summed.  Consequently, for ,

 L−1[exp(−E(η))η]ξ→1/a=1−ln(1a)+12[−π26+ln(a)2+2Li2(a)]

which is consistent with before.

## 2 Mutafchiev

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

 limn→∞P{Λ≤an}=1√aσ(1a),0

where, from the DDE,

 √xσ(x)=1−12ln⎛⎜ ⎜⎝1+√1−1x1−√1−1x⎞⎟ ⎟⎠if 1

For reasons of simplicity, change our example domain from to .  Hence

 limn→∞P{12<Λn≤23} =√32σ(32)−√2σ(2) =−12ln⎛⎜ ⎜⎝1+√1−231−√1−23⎞⎟ ⎟⎠+12ln⎛⎜ ⎜⎝1+√1−121−√1−12⎞⎟ ⎟⎠ =0.222894638557....

Let us again explore the less-familiar approach [13].  Define as previously.  From

 √πξL−1⎡⎢ ⎢⎣exp(−12E(η))√η⎤⎥ ⎥⎦=√πξ∞∑k=0(−1)k2kk!L−1[E(η)k√η]

we recognize two well-known formulas:

 L−1[1√η]=1√πξ,L−1[E(η)√η]=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩2√πξarctanh(√1−1ξ)if ξ≥1,0if 0≤ξ<1.

Since for , only terms of the series need to be summed.  Consequently, for ,

 √πaL−1⎡⎢ ⎢⎣exp(−12E(η))√η⎤⎥ ⎥⎦ξ→1/a=1−arctanh(√1−a)=1−12ln(1+√1−a1−√1−a)

which again is consistent with before.

As an aside, had we kept the domain as , then numerics are possible:

 limn→∞P{13<Λn≤12} =1/2∫1/312t3/2σ(1−tt)dt =121/2∫1/31t√1−t⎡⎢ ⎢⎣1−12ln⎛⎜ ⎜⎝1+√1−tt−11−√1−tt−1⎞⎟ ⎟⎠⎤⎥ ⎥⎦dt =0.110414874191....

but symbolics seem unlikely.  A closed-form expression for the inverse Laplace transform of also remains open.

## 3 Purdom & Williams

We change the subject to cycles, i.e., loops in the functional graph. Let denote the length of the longest cycle in an -mapping, chosen uniformly at random.  Our goal is to find .  While no relevant DDE is yet known, there is an associated inverse Laplace transform

due to Mutafchiev [13].  Unlike previously, holds (rather than ) and is the Laplace transform of .  Self-convolutions of this function do not enjoy the same vanishing properties as those for .  Truncating the infinite series, although it is convergent, will unfortunately lead to non-zero error.  An alternative expression

 √2π12πi1+i∞∫1−i∞exp(−E(bζ)+ζ22)dζ

is due to Stepanov [11].  Letting , i.e., and , demonstrates that this complex contour integral is equal to the preceding inverse Laplace transform.

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

the conditional probability of

given :

 P{Λ≤λ√n|N=ν√n}=ρ(νλ)

by [16, 17] the limiting density (as ) of :

 νexp(−ν22)(Rayleigh)%

and then differentiating (with respect to ) to obtain the joint density of :

 f(λ,ν) =νexp(−ν22)ρ′(νλ)(−νλ2) =νexp(−ν22)ρ(νλ−1)(−λν)(−νλ2) =νλexp(−ν22)ρ(ν−λλ)

where .  We have not seen this formula in the literature: it is apparently new.  For , let denote the length of the longest cycle in an -mapping, chosen uniformly at random.  If the permutation has no cycle, then its longest cycle is defined to have length .  Define the generalized Dickman function to satisfy

 ξρ′r(ξ)+ρr(ξ−1)=ρr−1(ξ−1) for ξ>1,ρr(ξ)=1 for 0≤ξ≤1

where .  By the same argument, for ,

Define

 Gr,h=1h!(r−1)!∞∫0xh−1E(x)r−1exp[−E(x)−x]dx

(in this paper, rank or ; height or ).  Purdom & Williams [18], building on Shepp & Lloyd [19]

, discovered asymptotic formulas for moments

.  We can easily verify their findings:

 limn→∞E(Λr)√n=√π2Gr,1=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩0.78248160099165661501...if r=1,0.26267067265131265469...if r=2,0.11068781528281010827...if r=3,0.05056118481134243184...if r=4;
 limn→∞V(Λr)n=2Gr,2−π2G2r,1=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩0.24111407342881901748...if r=1,0.04395998473216610374...if r=2,0.01233552055537805858...if r=3,0.00386619224804518754...if r=4.

The mode of occurs when

 0 =ddλ∞∫λf(λ,ν)dν=−f(λ,λ)+∞∫λ∂f∂λ(λ,ν)dν =−exp(−λ22)+∞∫λνexp(−ν22)∂∂λ[1λρ(νλ−1)]dν;

the inner derivative becomes

 −1λ2ρ(νλ−1)+1λρ′(νλ−1)(−νλ2) =−1λ2ρ(νλ−1)−1λρ(νλ−2)νλ−1(−νλ2) =−1λ2ρ(ν−λλ)+νλ2(ν−λ)ρ(ν−2λλ);

solving the equation

 exp(−λ22)=1λ2∞∫λ[−ρ(ν−λλ)+νν−λρ(ν−2λλ)]νexp(−ν22)dν

yields as the mode.  The median of arises simply from

 12=∞∫λ∞∫μf(μ,ν)dνdμ.

For , the mode is ; we did not pursue the median. Another new asymptotic result is the cross-correlation between and :

 limn→∞E(ΛrN)−E(Λr)E(N)√V(Λr)√V(N)=√2−π2Gr,1√2Gr,2−π2G2r,1=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩ccc0.83298010...if r=1,0.65486924...if r=2,0.52094617...if r=3,0.42505712...if r=4.

Using formulas in [10, 20, 21], it is possible to similarly compute the cross-correlation between and where .

## 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

 P{M=1}∼√π2nas n→∞

where counts the components.  Let denote the length of the unique cycle in a connected -mapping, chosen uniformly at random.  Our goal is to find as before, but the circumstances are vastly simpler.  Rényi [22] proved that the limiting density (as ) of  is

 √2πexp(−λ22)(half-normal)

for , which implies immediately that

 limn→∞E(Λ)√n=√2π=0.79788456080286535587...,
 limn→∞V(Λ)n=1−2π=0.36338022763241865692....

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

Of course, when there is just one component.  Allowing instead components, where is a fixed integer, rarity persists [23, 24]

 P{M=m}∼12m−1(m−1)!√π2nln(n)m−1as n→∞

but the asymptotic values and for and evidently do not change.  Section 6 contains more on this issue.

## 5 Pavlov

For arbitrary mappings, the expected number of components [25, 26] is .  If our constraint from Section 4 loosens so that but so slowly that , 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 be the longest cycle length and be the cyclic points total.  As before, a conditional probability coupled with the limiting density (as ) of :

 √2πexp(−ν22)

suffice to give the joint density of :

 f(λ,ν)=√2π1λexp(−ν22)ρ(ν−λλ)

where .  Further,

 fr(λ,ν)=√2π1λexp(−ν22)[ρr(ν−λλ)−ρr−1(ν−λλ)]

for .  Moments are

 limn→∞E(Λr)√n=√2πGr,1=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩0.49814325870512904597...if r=1,0.16722134383091813637...if r=2,0.07046605176920746245...if r=3,0.03218824996523203019...if r=4;
 limn→∞V(Λr)n=Gr,2−2πG2r,1=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩0.17854905846627743895...if r=1,0.02851495566901143371...if r=2,0.00732819205178914862...if r=3,0.00217522939296169629...if r=4.

The mode of occurs at ; the median at .  For , we did not pursue the median.  The cross-correlation between and is

 limn→∞E(ΛrN)−E(Λr)E(N)√V(Λr)√V(N)=√1−2πGr,1√Gr,2−2πG2r,1=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩0.89066843...if r=1,0.74816251...if r=2,0.62190221...if r=3,0.52141727...if r=4

and, again, it is possible to compute the cross-correlation between and where .

The mean is sharply less than the other means and we have exhibited.  Why should this counterintuitive fact be true?  The scenario is intermediate to the others.  This is why we describe our work here as conjectural.

If instead for some constant , then Pavlov’s [27] density formula is

 2cΓ(c)√2πΓ(2c)ν2cexp(−ν22).

This reduces to the density found in [16, 17] when .

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 denote the number of -mappings possessing exactly components and exactly cyclic points, where .  We have [33, 34, 35]

 ∞∑n=ℓn∑ℓ=manmℓn!xnyℓ=1m!ln(11−yτ(x))m

where is Cayley’s tree function.  The dominant singularity of is at and

 τ(x)∼1−21/2√1−exas x→e−1.

Differentiating with respect to :

 ∞∑n=ℓn∑ℓ=mℓanmℓn!xnyℓ−1=1(m−1)!τ(x)1−yτ(x)ln(11−yτ(x))m−1

and setting :

 1(m−1)!τ(x)1−τ(x)ln(11−τ(x))m−1 ∼1(m−1)!121/2√1−exln(121/2√1−ex)m−1 ∼12m−1/2(m−1)!√11−exln(11−ex)m−1

we deduce

 n∑ℓ=mℓanmℓn! ∼12m−1/2(m−1)!(1e)−n√nln(n)m−1nΓ(1/2) ∼ln(n)m−12m−1(m−1)!en√2πn

by the singularity analysis theorem of Flajolet & Odlyzko [36].  Multiplying both sides by and using Stirling’s approximation, we obtain

 n∑ℓ=mℓanmℓnn∼ln(n)m−12m−1(m−1)!.

From Section 4,

 n∑ℓ=manmℓnn∼ln(n)m−12m−1(m−1)!√π2n

and hence, forming the ratio, as .

Differentiating again and setting :

 ∞∑n=ℓn∑ℓ=mℓ(ℓ−1)anmℓn!xn ∼1(m−1)!(τ(x)1−τ(x))2ln(11−τ(x))m−1 ∼1(m−1)!12(1−ex)ln(121/2√1−ex)m−1 ∼12m(m−1)!11−exln(11−ex)m−1

we deduce

 n∑ℓ=mℓ(ℓ−1)anmℓn! ∼12m(m−1)!(1e)−nnln(n)m−1nΓ(1) ∼ln(n)m−12m(m−1)!en.

Multiplying by and via the preceding, we obtain

 n∑ℓ=mℓ2anmℓnn∼ln(n)m−12m(m−1)!√2πn∼ln(n)m−12m−1(m−1)!√πn2.

Forming the ratio, as and thus .

Let

be a positive integer.  A random variable

is -divisible if it can be written as , where are independent and identically distributed. A random variable is infinitely divisible if it is -divisible for every .  We wish to study the allocation of cyclic points among a fixed number of components, given a constrained random mapping.  This would be a matter of determining the inverse Laplace transform of the root of

 L[√2πexp(−ξ22)]=exp(η22)erfc(η√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 .  Let us offer a plausibility argument supporting this latter assertion.

On the one hand, if is fixed, then the density of clearly approaches as , i.e., it is bounded near the origin.

On the other hand, if no condition is placed on , then the density of is a convolution in the -domain, which becomes multiplication in the -domain.  Starting with [40, 41, 42]

 π√2π+πη<√π2exp(η22)erfc(η√2)<π√2π+2η

for all , we deduce

  ⎷11+√π2η

and upper/lower bounds are tight approximations of the center for small/large values of .  No closed-form expression for center seems to be possible; lower bound and upper bound are

 21/4π3/4√ξexp(−√2πξ)and1(2π)1/4√ξexp(−√π2ξ)

respectively.  Both expressions approach infinity as , tentatively implying that the density of is unbounded near the origin. This contrasts with the behavior described earlier, i.e., information about truly affects how is distributed.  Therefore and must be dependent.

When we employed the word “obstacle” before, it reflected our intention to study order statistics and , with a goal of understanding the allocation process (partioning cyclic points into two cycles).  If and were independent with common density , then the joint density of would simply be for .  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 .  We argue as before, but less formally.  On the one hand, if is fixed, then the density of approaches as .  On the other hand, if no condition is placed on , then we wish to find the inverse Laplace transform of the square root of

 L[ξexp(−ξ22)]=1−√π2ηexp(η22)erfc(η√2).

A remarkably accurate approximation (with error less than ):

  ⎷1−√π2ηexp(η22)erfc(η√2)≈exp(η2π)erfc(η√π)

defies easy explanation and yet provides a very helpful estimate:

 L−1[exp(η2π)erfc(η√π)]=exp(−πξ24)

which approaches as .  Since , it follows that and must be dependent.

With the benefit of hindsight, we should have focused not on , but instead on

 ~σ(x)=√xσ(x)

both here and in [10].  The derivative of is found as follows:

 ddx~σ(1x)=ddx[1x1/2σ(1x)]=−12x3/2σ(1x)+1x1/2σ′(1x)(−1x2)

but

 σ′(y)=−12y(σ(y)+σ(y−1))

therefore

 ddx~σ(1x) =−12x3/2σ(1x)+1x1/2(−x2)[σ(1x)+σ(1x−1)](−1x2) =−12x3/2σ(1x)+12x3/2[σ(1x)+σ(1x−1)]=12x3/2σ(1−xx),

as was to be shown.  Finally, and are subsumed by the general DDE [4, 5, 6]

 xg′(x)+(1−θ)g(x)+θg(x−1)=0 % for x>1,g(x)=xθ−1 for 0

where is fixed, and its associated Laplace transform is

 L[g(ξ)]=exp(−θE(η))ηθ/Γ(θ),η∈C∖(−∞,0].

It would be good someday to learn, from an interested reader, about possible random mapping-theoretic applications of  for select.

## 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 -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 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