DeepAI

# Permute, Graph, Map, Derange

We study decomposable combinatorial labeled structures in the exp-log class, specifically, two examples of type a=1 and two examples of type a=1/2. Our approach is to establish how well existing theory matches experimental data. For instance, the median length of the longest cycle in a random n-permutation is (0.6065...)*n, whereas the median length of the largest component in a random n-mapping is (0.7864...)*n. Unsolved problems are highlighted, in the hope that someone else might address these someday.

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## 1 Permute

For fixed , the sequences and constitute probability mass functions (upon normalization by )  These have corresponding means ,

and variances

, given in Table 1.  We also provide the median ; note that for is trivial.  For convenience (in table headings only), the following notation is used:

 L˜μn=Lμnn,L˜σ2n=Lσ2nn2,L˜νn=Lνnn,
 S˜μn=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩Sμnln(n)if a=1,Sμnn1/2if a=1/2;S˜σ2n=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩Sσ2nnif a=1,Sσ2nn3/2if a=1/2.
1000 0.624642 0.036945 0.6060 0.717352 1.307043
2000 0.624486 0.036926 0.6060 0.703135 1.307125
3000 0.624434 0.036920 0.6063 0.695960 1.307153
4000 0.624408 0.036917 0.6062 0.691295 1.307167

Table 1: Statistics for Permute ()

We have

 limn→∞Lμnn=LG1(1,1)=0.62432998854355087099...,
 limn→∞Lσ2nn2=LG1(1,2)−LG1(1,1)2=0.03690783006485220217...,
 limn→∞Lνnn=1√e=0.60653065971263342360...,
 limn→∞Sμnln(n)=e−γ=0.56145948356688516982...,
 limn→∞Sσ2nn=SG1(1,2)=1.30720779891056809974....

It is not surprising that  enjoys linear growth:  and jointly place considerable weight on the distributional extremes.  The unusual logarithmic growth of  is due to nevertheless overwhelming all other .

A one-line proof of the  result is [10, 11]

 limn→∞n∑k=⌊n√e⌋1k=limn→∞ln(n)−ln(n√e)=ln(1√e)=12.

Alternatively, the asymptotic probability that the longest cycle has size is 

 1∫x1ydy=12

hence and .  We will see a variation of this approach later.

No formula for the covariance between sizes of the longest cycle and shortest cycle is known.  Interplay between the number of cycles and either of the extremes likewise remains inscrutable.  We earlier examined not permutations, but instead integer compositions, finding a complicated recursion for a certain bivariate probability distribution

[12, 13]

.  Thus a cross-correlation can be estimated for compositions, but not yet for permutations

.

## 2 Graph

Let us explain why , ; , ; and , , .  When , a -regular graph is either a hexagon, with distinct labelings, or the disjoint union of two triangles, with labelings.  When , a -regular graph is either a heptagon, with distinct labelings, or the disjoint union of a triangle and a square, with labelings (since ).  When , a -regular graph is either an octagon, with distinct labelings; the disjoint union of a triangle and a pentagon, with labelings (since ); or the disjoint union of two squares, with labelings (since ).  Circumstances become more complicated when : a -regular graph is either an enneagon, or the disjoint union of a square and a pentagon, or the disjoint union of three triangles, or the disjoint union of a triangle and a hexagon .

Upon normalization by , we obtain

1000 0.758771 0.037099 0.7860 3.007677 2.097084
2000 0.758297 0.037053 0.7865 3.029960 2.096470
3000 0.758139 0.037038 0.7863 3.039930 2.096262
4000 0.758060 0.037030 0.7865 3.045902 2.096157

Table 2: Statistics for Graph ()

and

 limn→∞Lμnn=LG1/2(1,1)=0.75782301126849283774...,
 limn→∞Lσ2nn2=LG1/2(1,2)−LG1/2(1,1)2=0.03700721658229030320...,
 limn→∞Lνnn=4e(1+e)2=0.78644773296592741014...,
 limn→∞Sμnn1/2=e3/4SG1/2(1,1)=3.08504247563149222958...,
 limn→∞Sσ2nn3/2=e3/4SG1/2(1,2)=2.09583743942571712967....

Some non-explicit formulas for the latter two results arise in Section 7.  A proof for the median result is deferred to Section 3.

## 3 Map

Let us explain why , and , .  The unique -map with totally disconnected nodes is the identity map; we associate this map with its image sequence , i.e., two isolated loops (two components of size ).  The maps and are each pictured as one loop attached to a -tail (a component of size ); the map is pictured as a -cycle (again, a component of size ).  For -maps, we have

 123i.e., three isolated loops;
 113,121,122,133,223,323i.e., one isolated loop % and one loop attached to a 1-tail;
 213,321,132i.e., one isolated loop and one 2-cycle

which give ten cases, and

 112,131,221,322,233,313i.e., one loop attached to% a 2-tail;
 111,222,333i.e., one loop attached to two 1%−tails;
 211,212,232,311,331,332i.e., one 2-cycle % attached to a 1-tail;
 231,312i.e., one 3-cycle

which give seventeen cases [16, 17].  The complexity grows when : it can be shown that , , .

Upon normalization by , we obtain

1000 0.762505 0.036968 0.7920 1.969526 1.384968
2000 0.761122 0.036980 0.7905 1.991932 1.389355
3000 0.760512 0.036985 0.7900 2.002505 1.391309
4000 0.760149 0.036988 0.7895 2.009048 1.392477

Table 3: Statistics for Map ()

and

 limn→∞Lμnn=LG1/2(1,1)=0.75782301126849283774...,
 limn→∞Lσ2nn2=LG1/2(1,2)−LG1/2(1,1)2=0.03700721658229030320...,
 limn→∞Lνnn=4e(1+e)2=0.78644773296592741014...,
 limn→∞Sμnn1/2=√2SG1/2(1,1)=2.06089224152016653900...,
 limn→∞Sσ2nn3/2=√2SG1/2(1,2)=1.40007638550124502818....

As before, some non-explicit formulas for the latter two results arise in Section 7.  The asymptotic probability that the largest component has size is [18, 19]

 1∫x12y√1−ydy=12

hence

 12ln(1+√1−x1−√1−x)=12

and .

## 4 Derange

Derangements are permutations with no fixed points .  It is easy to show that (a longest cycle in a -derangement cannot have size or ) and .  Upon normalization by , we obtain

1000 0.625266 0.037018 0.6060 1.701217 3.551193
2000 0.624798 0.036963 0.6065 1.685257 3.552276
3000 0.624642 0.036945 0.6067 1.677202 3.552637
4000 0.624564 0.036935 0.6065 1.671965 3.552818

Table 4: Statistics for Derange ()

and

 limn→∞Lμnn=LG1(1,1)=0.62432998854355087099...,
 limn→∞Lσ2nn2=LG1(1,2)−LG1(1,1)2=0.03690783006485220217...,
 limn→∞Lνnn=1√e=0.60653065971263342360...,
 limn→∞Sμnln(n)=e1−γ=1.52620511159586388047...,
 limn→∞Sσ2nn=e⋅SG1(1,2)=3.55335920579854297440....

The asymptotic expression for the average shortest cycle length follows from

 Sμn⋅bn+1⋅(n!−bn)n!∼e−γln(n)

and the fact that as ; similarly for higher moments.

## 5 Generalized Dickman Rho (I)

Define to be the number of -objects whose largest component has size ; thus .  Given , let

 ρa(x)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1if 0≤x<1,1−ax∫1ρa(t−1)⋅(t−1)a−1tadtif x≥1

and observe that the standard Dickman function .  A theorem proved in  asserts that

for any .  Of course,

 m∑j=1Lj,n=bn,m

hence we can easily verify this result experimentally.

 m∖x 2 3 4 5 2 3 4 100 0.309347 0.049634 0.0050952 0.0003748 0.117715 0.0082644 0.0003680 200 0.308101 0.049121 0.0050026 0.0003646 0.118178 0.0082399 0.0003638 300 0.307685 0.048950 0.0049719 0.0003613 0.118329 0.0082309 0.0003624 400 0.307477 0.048864 0.0049566 0.0003597 0.118404 0.0082262 0.0003616 500 0.307353 0.048813 0.0049475 0.0003587 0.118449 0.0082233 0.0003612 600 0.307269 0.048779 0.0049414 0.0003580 0.118478 0.0082214 0.0003609 700 0.307210 0.048755 0.0049370 0.0003575 0.118500 0.0082200 0.0003607 800 0.307165 0.048736 0.0049337 0.0003572 0.118516 0.0082190 0.0003605 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∞ 0.306853 0.048608 0.0049109 0.0003547 0.118626 0.0082115 0.0003594

Table 5A: Ratio for Permute () and Graph ()

 m∖x 2 3 4 2 3 4 5 100 0.111305 0.0074576 0.0003185 0.304359 0.048597 0.0049699 0.0003645 200 0.112756 0.0076060 0.0003258 0.305604 0.048605 0.0049409 0.0003596 300 0.113579 0.0076901 0.0003302 0.306020 0.048607 0.0049310 0.0003580 400 0.114124 0.0077458 0.0003332 0.306228 0.048608 0.0049260 0.0003572 500 0.114518 0.0077862 0.0003354 0.306353 0.048608 0.0049230 0.0003567 600 0.114822 0.0078173 0.0003371 0.306436 0.048608 0.0049210 0.0003564 700 0.115065 0.0078423 0.0003384 0.306496 0.048608 0.0049196 0.0003561 800 0.115265 0.0078629 0.0003396 0.306540 0.048608 0.0049185 0.0003559 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∞ 0.118626 0.0082115 0.0003594 0.306853 0.048608 0.0049109 0.0003547

Table 5B: Ratio for Map () and Derange ()

Why have we devoted effort to evaluating Dickman’s rho?  Answer: the function is fundamentally connected to asymptotics in Sections 1–4.  The moments of the largest component size are

 ∞∫0ρ1(x)(x+1)h+1dxand12∞∫0ρ1/2(x)√x(x+1)h+1/2dx,h=1,2,…

for and , respectively.  Extension to arbitrary is possible.  Of course, we also have integrals available.

## 6 Generalized Buchstab Omega (I)

Define to be the number of -objects whose smallest component has size ; note that .  Given , let

 Ωa(x)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1if 1≤x<2,1+ax∫2Ωa(t−1)t−1dtif % x≥2

and observe that the standard Buchstab function .  A theorem proved in  asserts that

for any .  Of course,

 n∑j=mSj,n=bn,m

hence we can easily verify this result experimentally.

As an aside, is called the probability of connectedness in 

, i.e., the odds that an

-object, whose smallest component has size at least , is connected.  No analogous name has been proposed for from Section 5, i.e., the odds that all components of an -object have size at most .  Maybe probability of smoothness would be appropriate (“smooth” coming from prime number theory).  For Section 7, the same ratio might be called the probability of roughness, wherein all components of an -object have size at least .

 m∖x 2 3 4 5 2 3 4 5 100 0.990 0.587992 0.443034 0.354438 0.995 0.740555 0.628689 0.555092 200 0.995 0.589306 0.444151 0.355327 0.997 0.741591 0.629581 0.555860 300 0.997 0.589743 0.444523 0.355624 0.998 0.741936 0.629878 0.556116 400 0.997 0.589962 0.444710 0.355772 0.999 0.742108 0.630027 0.556244 500 0.998 0.590093 0.444822 0.355862 0.999 0.742212 0.630116 0.556321 600 0.998 0.590180 0.444896 0.355921 0.999 0.742281 0.630175 0.556372 700 0.999 0.590242 0.444949 0.355963 0.999 0.742330 0.630218 0.556409 800 0.999 0.590289 0.444989 0.355995 0.999 0.742367 0.630250 0.556436 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∞ 1 0.590616 0.445269 0.356218 1 0.742626 0.630473 0.556628

Table 6A: Ratio for Permute () and Graph ()

 m∖x 2 3 4 5 2 3 4 5 100 0.995 0.746112 0.635215 0.561960 0.990 0.587992 0.443034 0.354438 200 0.998 0.745502 0.634175 0.560698 0.995 0.589306 0.444151 0.355327 300 0.998 0.745124 0.633623 0.560060 0.997 0.589743 0.444523 0.355624 400 0.999 0.744866 0.633267 0.559656 0.997 0.589962 0.444710 0.355772 500 0.999 0.744677 0.633013 0.559371 0.998 0.590093 0.444822 0.355862 600 0.999 0.744530 0.632819 0.559156 0.998 0.590180 0.444896 0.355921 700 0.999 0.744412 0.632664 0.558985 0.999 0.590242 0.444949 0.355963 800 0.999 0.744314 0.632537 0.558845 0.999 0.590289 0.444989 0.355995 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∞ 1 0.742626 0.630473 0.556628 1 0.590616 0.445269 0.356218

Table 6B: Ratio for Map () and Derange ()

Buchstab’s Omega, as defined here, does not seem to be allied with asymptotics in Sections 1–4.  A different generalization is discussed in Section 7.

## 7 Generalized Buchstab Omega (II)

Define to be the number of -objects whose smallest component has size (as in Section 6).  When restricting attention to permutations, Panario & Richmond  obtained that

 limm→∞mb⌊xm⌋,mb⌊xm⌋=ω(x)

for any , where is the standard Buchstab function.  They seemed to presume that the same limit would occur for derangements (since both permutations and derangements have ), which is not true.  Replace now the initial factor in the numerator by .  Panario & Richmond realized that -regular graphs and mappings would possess a limit different from .  They seemed, however, to presume that equivalent limits would occur (since both graphs and maps have ), which is again untrue.  In Section 5, we studied two functions , ; here we have four omega (lowercase “o”) functions , , one for each structure under consideration.  Upon multiplication of limits, we discover

 ωA(x)Ωa(x)=limm→∞mac⌊xm⌋b⌊xm⌋=1xa⋅⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1if A=P and a=1,e3/4√π/2if A=G and a=1/2,√π/2if A=M and a=1/2,eif A=D and a=1

using known asymptotics as for graphs and mappings [23, 24, 25].  Perhaps, for fixed , varies only up to multiplicative constant.  These formulas allow us to provide numerical values in the final rows of Tables 6A and 6B.

Return now to Panario & Richmond. Especially puzzling is a claim (for permutations) that 

 limn→∞Sσ2nn=∞∫2ω(x)x2dx.

From Section 1, the left-hand side is  whereas the right-hand side is .  Thus predictions in  for are evidently mistaken.

m∖x 100 200 300 400 500 600 700 2 3 4 5 2 3 4 5 0.50500 0.56690 0.56429 0.56427 1.33744 1.46573 1.49445 1.51342 0.50250 0.56564 0.56287 0.56286 1.33203 1.46216 1.49116 1.51038 0.50166 0.56522 0.56240 0.56239 1.33023 1.46097 1.49007 1.50937 0.50125 0.56501 0.56216 0.56216 1.32933 1.46037 1.48952 1.50887 0.50100 0.56488 0.56202 0.56202 1.32879 1.46002 1.48920 1.50856 0.50083 0.56480 0.56193 0.56192 1.32843 1.45978 1.48898 1.50836 0.50072 0.56474 0.56186 0.56186 1.32817 1.45961 1.48882 1.50822 0.50063 0.56470 0.56181 0.56181 1.32798 1.45948 1.48871 1.50811 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0.5 0.56438 0.56146 0.56145 1.32663 1.45860 1.48789 1.50735

Table 7A: Ratio for Permute () and Graph ()

m∖x 100 200 300 400 500 600 700 2 3 4 5 2 3 4 5 0.87413 0.95520 0.97361 0.98570 1.37273 1.54100 1.53390 1.53385 0.87676 0.96022 0.97895 0.99132 1.36594 1.53756 1.53005 1.53001 0.87816 0.96259 0.98148 0.99397 1.36367 1.53642 1.52876 1.52874 0.87906 0.96406 0.98303 0.99559 1.36254 1.53585 1.52812 1.52810 0.87971 0.96508 0.98411 0.99672 1.36186 1.53551 1.52774 1.52772 0.88021 0.96584 0.98492 0.99756 1.36141 1.53528 1.52748 1.52746 0.88060 0.96644 0.98555 0.99823 1.36109 1.53512 1.52730 1.52728 0.88092 0.96693 0.98607 0.99876 1.36084 1.53500 1.52716 1.52715 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0.88623 0.97438 0.99395 1.00695 1.35914 1.53415 1.52620 1.52619

Table 7B: Ratio for Map () and Derange ()

Why have we devoted effort to evaluating Buchstab’s omega?  Answer: an array of formulas, parallel to those involving , corresponding to moments of the smallest component size, were proposed in :

 ∞∫2ωA(x)xh+1/2dx,h=1,2,…,for A∈{G,M}

and would be exceedingly attractive.  Unfortunately the potential for fulfillment is not good.  No high-precision numerical estimates of these integrals are currently known; thus we are not certain that any of the various are necessarily allied with asymptotics in Sections 1–4.  For now, the formulas remain frustratingly non-explicit and unverified.

## 8 Generalized Dickman Rho (II)

Define to be the number of -objects whose largest component has size (as in Section 5).  When restricting attention to permutations, we observe that

 limm→∞⌊xm⌋c⌊xm⌋b⌊xm⌋,m=1ρ(x)

for any , where is the standard Dickman function.  When restricting attention to derangements, a factor of needs to be included (just as in Sections 4 and 7).  Replace now the initial factor in the numerator by .  Again -regular graphs and mappings possess non-equivalent limits different from .  Just as we found the defining limit for Buchstab’s omega in terms of earlier, here we discover the limit in terms of :

We know that is important (Section 5) and believe that deserves further study (Section 7).  It is hoped that someone else might succeed in carrying on research where we have stopped.

 m∖x 2 3 4 5 2 3 4 5 100 3.23262 20.1473 196.264 2668.39 15.9879 227.490 5106.02 161434. 200 3.24569 20.3580 199.898 2742.41 15.9005 227.928 5161.08 164830. 300 3.25007 20.4291 201.130 2767.66 15.8719 228.098 5180.23 166005. 400 3.25227 20.4648 201.750 2780.41 15.8577 228.188 5189.95 166600. 500 3.25359 20.4863 202.124 2788.09 15.8492 228.244 5195.83 166961. 600 3.25447 20.5006 202.373 2793.22 15.8436 228.281 5199.78 167202. 700 3.25510 20.5109 202.552 2796.90 15.8396 228.308 5202.60 167375. 800 3.25558 20.5186 202.686 2799.66 15.8365 228.329 5204.73 167504. ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∞ 3.25889 20.5726 203.628 2819.09 15.8156 228.476 5219.73 168421.

Table 8A: Ratio for Permute () and Graph ()

 m∖x 2 3 4 5 2 3 4 5 100 11.0531 165.524 3883.12 128134. 8.93117 55.9354 546.943 7458.53 200 10.9698 163.012 3811.16 125549. 8.89477 55.9254 550.162 7558.92 300 10.9164 161.547 3766.89 123839. 8.88269 55.9236 551.265 7593.21 400 10.8799 160.574 3737.04 122663. 8.87665 55.9229 551.822 7610.52 500 10.8531 159.869 3715.21 121794. 8.87304 55.9226 552.159 7620.95 600 10.8322 159.327 3698.37 121120. 8.87063 55.9224 552.384 7627.93 700 10.8155 158.894 3684.85 120577. 8.86890 55.9223 552.545 7632.92 800 10.8016 158.537 3673.69 120127. 8.86761 55.9223 552.665 7636.67 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∞ 10.5652 152.628 3486.92 112510. 8.85859 55.9221 553.517 7663.07

Table 8B: Ratio for Map () and Derange ()

At the conclusion of Section 5, we gave expressions for the moments of largest component size, given or , without justification.  Here is a plausibility argument. Assuming (absent any proof) that the first moment for arbitrary is 

 λa=1−∞∫1ρa(x)x2dx,

we reverse integration-by-parts:

and obtain

i.e.,

From here, we infer (again, absent any proof) that the moment is

An explanation for and asymptotics in Sections 2 and 3 was not given.  Reason: it is more complicated than the proof in Section 4.  At some later point, we hope to study combinatorial objects called cyclations  for which moments are known to be precisely .  Since as for these, and because corresponding limits for graphs and mappings appear in Section 7, the factors and emerge.

## 10 Acknowledgements

I am grateful to Nicholas Pippenger  for a helpful discussion.  The creators of Mathematica, as well as administrators of the MIT Engaging Cluster, earn my gratitude every day.  A sequel to this paper is now available ; another will be released soon .

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