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Rounds, Color, Parity, Squares

This is a sequel to our paper "Permute, Graph, Map, Derange", involving decomposable combinatorial labeled structures in the exp-log class of type a=1/2, 1, 3/2, 2. As before, our approach is to establish how well existing theory matches experimental data and to raise open questions.

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11/10/2021

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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 Cycle-Colored 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 Cycle-Colored 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 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

Similarly,

yields

although the rightmost column of Table 2B suggests that this approximation is poor.

3 Component-Colored 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 Component-Colored 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 “post-processing removal” is acceptable (rather than a “pre-processing 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 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

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 exp-log 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

I am grateful to Mike Spivey [22], Alois Heinz and Michael Somos for helpful discussions.  The creators of Mathematica, as well as administrators of the MIT Engaging Cluster, earn my gratitude every day.  A sequel to this paper will be released soon [23].

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    Steven Finch
    MIT Sloan School of Management
    Cambridge, MA, USA
    steven_finch@harvard.edu