[seqfan] Re: Sequence related to jittery function

[bacher]

Hi,

I believe that something interesting is going on with
Jens Voss's bijections:

Every orbit among the n(n-1)/2 elements of X_n gives rise to a cyclic pattern
by considering \lfloor (a_i+b_i)/n\rfloor in {0,1} along the orbit.

Not every pattern is possible (this excludes the case of orbits
of length 2 but it should be possible to characterize such orbits).

A given cyclic pattern occurs then in an arithmetic progression of
possible values for n.

This leads to perhaps interesting identities summing
since X_n is the union of all its orbits.

It would be nice to work this out. I will think I
will give it a try.

Best wishes, Roland Bacher


bacher <Roland.Bacher at univ-grenoble-alpes.fr> a écrit :

> Sorry,
>
> I should have thought things out a bit longer:
> x=-1/2 is trivial since 2n+1=0 for $n=x=-1/2
> and the second transformation is the almost equal
> to the first (there remains some work to do but
> I should pan out).
>
>
> bacher <Roland.Bacher at univ-grenoble-alpes.fr> a écrit :
>
>> Hi Jens,
>> Try
>>
>> (a,b)=(38731904,40833784) for n=67108863
>>
>> for k=27
>>
>> (found randomly as follows:
>>
>> Consider a,b and x=n as variables and set up a cycle
>> involving 27 of your two transformation different from 27
>> iterations of the first formula and apply it to (a,b)
>> getting (l_1,l_2) with l_1,l_2 two linear forms in a,b,x + constants.
>> Solve the system for a=l_1,b=l_2 getting affine expressions
>> with rational coefficients in terms of x.
>> It seems that x=-1/2 is always a solution to both
>> systems (I tried a few times with random compositions, this seems  
>> also not to depend on the cycle length 27)
>> Computing -1/2 modulo the numerator of the rational expression for  
>> a or b yields n,
>> plugging this values in the expressions for a and b, yields a,b.
>>
>> Check that it works using your formulae (it did with my solution,
>> I hope I did not misstype it).
>>
>> I did not really think this over: all these things should be rather trivial
>> to prove or this has to be a wonderful theory (I suspect that the
>> first assertion is true)
>>
>> Best wishes Roland Bacher
>>
>> Jens Voß <jens at voss-ahrensburg.de> a écrit :
>>
>>> Let n be an integer greater than or equal to 2, and define
>>>
>>>   X_n := { (a, b) | 1 <= a < b <= n }
>>>
>>> to be the set of ordered pairs of distinct positive integers <= n.
>>>
>>> For every such pair (a, b) € X_n, let
>>>
>>>                  /  (b-a, b+a)      if  b+a <= n
>>>   j_n ((a,b)) := |
>>>                  \  (b-a, 2n+1-b-a) otherwise
>>>
>>> It is easy to see that
>>>
>>> (I)  j_n ((a,b)) € X_n for all (a, b) € X_n, i.e. j_n: X_n -> X_n and
>>> (II) j_n is injective.
>>>
>>> Since X_n is finite, j_n is bijective, i.e. a permutation of X_n.
>>>
>>> I am interested in the cycle representation of j_n; in particular,
>>> for every positive integer k, I would like to know the set of values
>>> of n for which j_n contains a cycle of length k.
>>>
>>> It is not too hard to show that
>>>
>>> (1) j_n has a cycle of length 1 (i.e. a fixed point) iff n == 2 (mod 5).
>>> (2) No n exists for which j_n has a cycle of length 2.
>>> (3) j_n has a cycle of length 3 iff n == 3 (mod 7) or n == 6 (mod 13).
>>>
>>> I suppose that larger cycle lengths can be classified in a similar way,
>>> but calculations quickly become rather tedious, so I used a simple
>>> computer program to calculate the cycle lengths of j_n for all values
>>> of n up to 3000 (after that, the program gets really slow).
>>>
>>> It appears that most smaller cycle lengths (except of course for 2) do
>>> come up eventually, the first stubborn exception being the number 27.
>>> However, the fact that this number (as well as the subsequent exceptions
>>> 38, 41, 53, 57, 62, 98, 103, 122, ...) does not show up as a cycle
>>> length of an j_n with n <= 3000 does not necessarily mean it can't do so
>>> for some larger n.
>>>
>>> So the questions I am facing right now are: Is 27 a cycle length of an
>>> f_n for some n? If so, what is the smallest such n? How about the other
>>> exceptions I mentioned? How many numbers k > 2 do not occur as cycle
>>> lengths; are there infinitely or finitely many or none at all?
>>>
>>> Does anyone have an idea on how to tackle these kind of problems? Has the
>>> function j_n perhaps already been studied by someone - it does not look
>>> too artificial to me!
>>> Ideally, if every integer > 2 is in fact a cycle length of some j_n, I'd
>>> like to submit the sequence of the smallest such n - currently, I am
>>> limited to the values for 3 through 26 and don't even know whether it
>>> can be extended beyond that point or whether it has a "hole" at k=27.
>>>
>>> Thanks and best regards
>>> Jens
>>>
>>>
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
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>
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