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:
Table 1: Statistics for Permute ()
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 .
Alternatively, the asymptotic probability that the longest cycle has size is 
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.
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
Table 2: Statistics for Graph ()
Some non-explicit formulas for the latter two results arise in Section 7. A proof for the median result is deferred to Section 3.
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
Upon normalization by , we obtain
Table 3: Statistics for Map ()
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
Table 4: Statistics for Derange ()
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  asserts that
for any . Of course,
hence we can easily verify this result experimentally.
Table 5A: Ratio for Permute () and Graph ()
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  asserts that
for any . Of course,
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 .
Table 6A: Ratio for Permute () and Graph ()
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
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 
Table 7A: Ratio for Permute () and Graph ()
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 :
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
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.
Table 8A: Ratio for Permute () and Graph ()
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 
we reverse integration-by-parts:
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.
-  S. W. Golomb and P. Gaal, On the number of permutations of objects with greatest cycle length , Adv. in Appl. Math. 20 (1998) 98–107; MR1488234.
-  D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica 31 (2001) 413–432; MR1855258.
-  D. Panario and B. Richmond, Smallest components in decomposable structures: exp-log class, Algorithmica 29 (2001) 205–226; MR1887304.
-  L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966) 340–357; MR0195117.
-  T. Shi, Cycle lengths of -biased random permutations, B.S. thesis, Harvey Mudd College, 2014, http://scholarship.claremont.edu/hmc_theses/65/.
-  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.
-  X. Gourdon, Combinatoire, Algorithmique et Géométrie des Polynômes, Ph.D. thesis, École Polytechnique, 1996.
-  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.
-  R. G. Pinsky, A view from the bridge spanning combinatorics and probability, arXiv:2105.13834.
-  J. Baez, Random permutations (Part 4), http://math.ucr.edu/home/baez/permutations/.
-  Wikipedia contributors, Random permutation statistics, Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Random_permutation_statistics.
-  M. Nej and A. Satyanarayana Reddy, Binary strings of length with zeros and longest -runs of zeros, Indian J. Math. 61 (2019) 111–139; arXiv:1707.02187; MR3931610.
-  S. R. Finch, Covariance within random integer compositions, arXiv:2010.06643.
-  N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A008275, A126074, and A145877.
-  N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A001205, A201013, A348070, and A348071.
-  R. Sedgewick and P. Flajolet, Introduction to the Analysis of Algorithms, Addison-Wesley, 1996, p. 460.
-  N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A060281, A209324, and A347999.
-  V. F. Kolchin, A problem of the allocation of particles in cells and cycles of random permutations, Theory Probab. Appl. 16 (1971) 74–90; MR0283840.
-  P. J. Donnelly, W. J. Ewens and S. Padmadisastra, Functionals of random mappings: exact and asymptotic results, Adv. in Appl. Probab. 23 (1991) 437–455; MR1122869.
-  N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A000166, A008306, A211871, and A348075.
-  M. Omar, D. Panario, B. Richmond and J. Whitely, Asymptotics of largest components in combinatorial structures, Algorithmica 46 (2006) 493–503; MR2291966.
-  E. A. Bender, A. Mashatan, D. Panario and L. B. Richmond, Asymptotics of combinatorial structures with large smallest component, J. Combin. Theory Ser. A 107 (2004) 117–125; MR2063956.
-  L. Katz, Probability of indecomposability of a random mapping function, Annals Math. Statist. 26 (1955) 512–517; MR0070869.
-  L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., 1974, pp. 273–278; MR0460128.
-  N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A001205, A001710, and A001865.
-  D. Panario and B. Richmond, Analysis of Ben-Or’s polynomial irreducibility test, Random Structures Algorithms 13 (1998) 439–456; MR1662794.
-  J. C. Lagarias, Euler’s constant: Euler’s work and modern developments, Bull. Amer. Math. Soc. 50 (2013) 527–628; MR3090422.
-  N. Pippenger, Random cyclations, Elec. J. Combin. 20 (2013) R9; arXiv:math/0408031; MR3139394.
-  S. R. Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487.
S. R. Finch, Second best, Third worst, Fourth in line,
Steven Finch MIT Sloan School of Management Cambridge, MA, USA email@example.com