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: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
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 oneline proof of the result is [10, 11]
Alternatively, the asymptotic probability that the longest cycle has size is [1]
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 crosscorrelation can be estimated for compositions, but not yet for permutations
[14].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 [15].
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
Some nonexplicit 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
which give ten cases, and
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
4 Derange
Derangements are permutations with no fixed points [20]. 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
The asymptotic expression for the average shortest cycle length follows from
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
and observe that the standard Dickman function . A theorem proved in [21] asserts that
for any . Of course,
hence we can easily verify this result experimentally.
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 ()
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
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
and observe that the standard Buchstab function . A theorem proved in [22] asserts that
for any . Of course,
hence we can easily verify this result experimentally.
As an aside, is called the probability of connectedness in [22]
, 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 .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 ()
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 [3] obtained that
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
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 [3]
From Section 1, the lefthand side is [4] whereas the righthand side is [26]. Thus predictions in [3] for are evidently mistaken.
2  3  4  5  2  3  4  5  
100  0.50500  0.56690  0.56429  0.56427  1.33744  1.46573  1.49445  1.51342 
200  0.50250  0.56564  0.56287  0.56286  1.33203  1.46216  1.49116  1.51038 
300  0.50166  0.56522  0.56240  0.56239  1.33023  1.46097  1.49007  1.50937 
400  0.50125  0.56501  0.56216  0.56216  1.32933  1.46037  1.48952  1.50887 
500  0.50100  0.56488  0.56202  0.56202  1.32879  1.46002  1.48920  1.50856 
600  0.50083  0.56480  0.56193  0.56192  1.32843  1.45978  1.48898  1.50836 
700  0.50072  0.56474  0.56186  0.56186  1.32817  1.45961  1.48882  1.50822 
800  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 ()
2  3  4  5  2  3  4  5  
100  0.87413  0.95520  0.97361  0.98570  1.37273  1.54100  1.53390  1.53385 
200  0.87676  0.96022  0.97895  0.99132  1.36594  1.53756  1.53005  1.53001 
300  0.87816  0.96259  0.98148  0.99397  1.36367  1.53642  1.52876  1.52874 
400  0.87906  0.96406  0.98303  0.99559  1.36254  1.53585  1.52812  1.52810 
500  0.87971  0.96508  0.98411  0.99672  1.36186  1.53551  1.52774  1.52772 
600  0.88021  0.96584  0.98492  0.99756  1.36141  1.53528  1.52748  1.52746 
700  0.88060  0.96644  0.98555  0.99823  1.36109  1.53512  1.52730  1.52728 
800  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 [3]:
and would be exceedingly attractive. Unfortunately the potential for fulfillment is not good. No highprecision 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 nonexplicit 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
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 nonequivalent 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.
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 ()
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 ()
9 Addendum
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 [27]
we reverse integrationbyparts:
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 [28] 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
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S. R. Finch, Second best, Third worst, Fourth in line,
forthcoming.
Steven Finch MIT Sloan School of Management Cambridge, MA, USA steven_finch@harvard.edu