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 ,
Table 1A: Statistics for Children’s Rounds ()
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 .
If we disallow the outer ring from having just one child, then clearly , and are all zero. We obtain
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 .
2 Cycle-Colored Permutations
Assuming two colors are available , 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
Table 2A: Statistics for Cycle-Colored Permutations ()
No explicit integral is known for the constant – it appears again shortly – the limit is proved in Theorem 5 of .
If we prohibit -cycles, then clearly , and are all zero; however We obtain
Table 2B: Statistics for Cycle-Colored 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
although the rightmost column of Table 2B suggests that this approximation is poor.
3 Component-Colored Mappings
Assuming three colors are available , it is clear that
An argument in Section 3 of  gives , , and . Similarly, , , , , and .
Upon normalization by , we obtain
Table 3: Statistics for Component-Colored Mappings ()
No explicit integrals are known for the latter two results.
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 “post-processing removal” is acceptable (rather than a “pre-processing removal”), then for the remaining components we easily have 
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 integer-based 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]. Clearly
holds 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
Table 4A: Statistics for EV permutations ()
Table 4B: Statistics for OD permutations (, )
Table 4C: Statistics for OD permutations (, )
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 exp-log parameter (numerical bounds, if relevant) and corresponding statistics.
We justify two of the limiting median formulas. For ,
when , i.e., , i.e., . For
when ; this is true because
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Steven Finch MIT Sloan School of Management Cambridge, MA, USA email@example.com