Two ordering problems

[Max Alekseyev]

At first glance, this property is also only necessary but not
sufficient (unless proved otherwise), since in David's sequence the
coefficients a and c are related to each other. Hence, a sequence
generated by floor(a*n+c) for some (arbitrary) a and c may not
necessary be possible to generate in a way proposed by David.

Regards,
Max

On Wed, Aug 27, 2008 at 3:59 PM,  <franktaw at netscape.net> wrote:
> The property you need is similar to that for a Beatty sequence.  The
> position
> of the nth X (or Y) in the sequence will be floor(a*n+c) for some a and c.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: David Wilson <dwilson at gambitcomm.com>
>
> Max Alekseyev wrote:
>>
>> It appears that the sequence is good iff the number of Y (resp. X)
>> symbols between any two neighboring X (resp. Y) symbols either equals
>> an integer constant or varies between some two consecutive integer
>> values.
>> In the aforementioned sample sequence
>> X,Y,X,Y,X,X,Y,X,Y,X,X,Y,...
>> the distance between every two neighboring X's is 0 or 1 and the
>> distance between every two neighboring Y's is 1 or 2.
>>
>> With this characterization in mind, it is easy to compute the number
>> of good sequences of length n.
>>
> Your observation is certainly a property of good sequences, indeed the
> distance between any two adjacent X's will always be k or k+1 for some k
> (similarly for adjacent Y's). This is necessary for a good sequence, but
> not sufficient. For example
>
> X,Y,X,Y,X,Y,X,X,Y,X,X
>
> is not good.
>