1 Children’s Rounds
Consider the myriad arrangements of labeled children into rounds [12, 13, 14]. A round means the same as a directed ring or circle, with exactly one child inside the ring and the others encircling. We permit the outer ring to have as few as one child. For , there are ways to make one round (a round of children, inside and outside):
and ways to make two rounds (both rounds):
implying that and . For , there are ways to make a round and ways to make a round & a round, implying that and . For , there are ways to make a round, ways to make a round & a round, ways to make two rounds, and ways to make three rounds, implying that , , , and . We obtain
and, upon normalization by ,
1000  0.621184  0.036672  0.6020  0.862134  1.317448 

2000  0.622539  0.036764  0.6040  0.834706  1.312715 
3000  0.623052  0.036802  0.6050  0.820866  1.311031 
4000  0.623326  0.036823  0.6053  0.811867  1.310156 
Table 1A: Statistics for Children’s Rounds ()
also
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 .
1.1 Variant
If we disallow the outer ring from having just one child, then clearly , and are all zero. We obtain
1000  0.622431  0.036818  0.6030  1.846026  3.574315 

2000  0.623163  0.036837  0.6050  1.816840  3.564892 
3000  0.623468  0.036851  0.6053  1.802115  3.561459 
4000  0.623638  0.036860  0.6055  1.792542  3.559653 
Table 1B: Statistics for Variant of Children’s Rounds ()
and, while the limits are the same as before, the limits differ by a factor of :
as argued in Section 4 of [1].
2 CycleColored Permutations
Assuming two colors are available [15], it is clear that
Let us explain why and . The permutations
each have possible colorings, giving to both and ; the permutation contributes another to . In contrast, , and because the permutations
again give to and , whereas the permutations
give to ; also, the permutations
contribute another to .
Upon normalization by , we obtain
1000  0.476115  0.027160  0.4480  1.292899  1.011228 

2000  0.475877  0.027132  0.4480  1.291149  0.976960 
3000  0.475798  0.027123  0.4480  1.290504  0.959564 
4000  0.475758  0.027119  0.4480  1.290163  0.948225 
Table 2A: Statistics for CycleColored Permutations ()
and
No explicit integral is known for the constant – it appears again shortly – the limit is proved in Theorem 5 of [4].
2.1 Variant
If we prohibit cycles, then clearly , and are all zero; however We obtain
1000  0.477065  0.027268  0.4480  3.159931  6.534345 

2000  0.476353  0.027187  0.4485  3.149165  6.372009 
3000  0.476115  0.027160  0.4483  3.145123  6.288169 
4000  0.475996  0.027146  0.4482  3.142963  6.233112 
Table 2B: Statistics for CycleColored Derangements ()
and, while the limits are the same as before, the limits differ somewhat:
The expresson involving for the average shortest cycle length follows from
and the fact that as ; hence
Similarly,
yields
although the rightmost column of Table 2B suggests that this approximation is poor.
3 ComponentColored Mappings
Assuming three colors are available [16], it is clear that
An argument in Section 3 of [1] gives , , and . Similarly, , , , , and .
Upon normalization by , we obtain
1000  0.544944  0.032583  0.5170  2.590160  5.925381 

2000  0.542744  0.032407  0.5155  2.597466  6.079830 
3000  0.541778  0.032331  0.5150  2.600326  6.152861 
4000  0.541205  0.032285  0.5142  2.601910  6.198088 
Table 3: Statistics for ComponentColored Mappings ()
and
No explicit integrals are known for the latter two results.
3.1 Variant
Instead of removing the smallest possible components (components), we wonder about removing the largest possible components. Of course, this can be done only imprecisely, as the size of the giant component cannot be known beforehand. If a “postprocessing removal” is acceptable (rather than a “preprocessing removal”), then for the remaining components we easily have [8]
Both mean and variance are significantly reduced (from to and to , respectively).
The recursive formulas for and , however, apply expressly when the desired rank . We cannot use our current exact integerbased algorithm to experimentally confirm these statistics for . A Monte Carlo simulation would be feasible, but less thorough and less accurate.
4 Parity of Cycle Lengths
Let EV and OD refer to permutations with all cycle lengths even and with all cycle lengths odd, respectively
[17, 18, 19]. Clearlyholds for EV permutations and
holds for OD permutations. On the one hand, in EV we have for all and odd . On the other hand, in OD we have always (due to the identity permutation) and , (both quickly checked). The fact that for even but for odd gives the divergence between Tables 4B & 4C for OD.
Upon normalization by , we obtain
1000  0.758202  0.037044  0.7850  2.028405  1.400424 

2000  0.758012  0.037026  0.7865  2.037816  1.400250 
3000  0.757949  0.037020  0.7863  2.042013  1.400192 
4000  0.757918  0.037016  0.7862  2.044523  1.400163 
Table 4A: Statistics for EV permutations ()
1000  0.757601  0.036937  0.7860  0.552117  0.125460 

2000  0.757712  0.036972  0.7860  0.553094  0.125440 
3000  0.757749  0.036984  0.7860  0.553535  0.125434 
4000  0.757768  0.036990  0.7860  0.553800  0.125431 
Table 4B: Statistics for OD permutations (, )
999  0.758045  0.037077  0.7845  1.501395  1.274342 

1999  0.757934  0.037042  0.7864  1.502628  1.274497 
2999  0.757897  0.037031  0.7863  1.503154  1.274549 
3999  0.757878  0.037025  0.7862  1.503461  1.274575 
Table 4C: Statistics for OD permutations (, )
and
As before, no explicit integrals are known for the latter four results.
5 Square Permutations
A permutation is a square () if and only if, for any integer , the number of cycles of length (in its disjoint cycle decomposition) must be even [17, 20, 21]. There is no restriction on the number of cycles of length . Hence both
are squares, but
are not squares. While it is not possible to enforce restrictions on cycle counts using the values of alone, we can still employ brute force methods to calculate
A significantly faster algorithm might provide insight on the explog parameter (numerical bounds, if relevant) and corresponding statistics.
6 Addendum
We justify two of the limiting median formulas. For ,
when , i.e., , i.e., . For
when ; this is true because
7 Acknowledgements
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[23]
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