[seqfan] Re: Evaluating sequences by same terms and different indices

[Bob Selcoe]

Hi Frank,

Thanks for the reply.  The issue isn't how to show the indexing; it's to 
describe the comparisons.

For instance, A064413 and A255582.  What would f and g be that would express 
their similarity?  What are the "explicit formulas"??

Bob S

--------------------------------------------------
From: "Frank Adams-Watters" <franktaw at netscape.net>
Sent: Thursday, March 05, 2015 7:34 PM
To: <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Evaluating sequences by same terms and 
differentindices

> I don't quite see the problem. You can enter a formula like:
>
> a(f(n)) = b(n) or a(n) = b(g(n)) or even a(f(n)) = b(g(n))
>
> (where f and g are explicit formulas, and b is an A-number).
>
> What further description do you need?
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Bob Selcoe <rselcoe at entouchonline.net>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Thu, Mar 5, 2015 7:22 pm
> Subject: [seqfan] Re: Evaluating sequences by same terms and 
> differentindices
>
>
> Hi Olivier,
>
> Thanks so much for the input (I'll look up the terms with which I'm
>
> unfamiliar), but perhaps I was unclear.  I wasn't really interested in the
>
> nature of permutations, per se.
>
> Rather, I'm trying to find a good way to
> describe comparing sequences
> A,B,C..,Y by their different indices where their
> terms are the same.  So
> a(f) = b(g) = c(h).. = y(z), but f, g, h.., z may not
> be equal.
>
> This requires that the different sequences share all of the same
> terms (i.e.
> are permutations of each other): they must in order to compare the
> all of
> the indices.
>
> Contrast this with comparing sequences by their same
> indices where their
> terms may be different (the way formulas often are
> presented in OEIS, for
> instance).  That is, f = g = h.. = z, but a(f), b(g),
> c(h).., y(z) may not
> be equal. Here, there is no need for the sequences to
> share all of the same
> terms.
>
> This is the general concept.  I hope it's clearer
> now.
>
> Specifically, though, I'm interested simply in comparing A064413 (the EKG
>
> sequence) and A255582, which are very similar in growth and behavior. 
> Both
>
> are permutations of the natural numbers.
>
> I'm trying come up with a way to
> describe this idea:
>
> A064413(m) = B255582(n) when m~n.   This is one way of
> saying the two
> sequences are "similar".
>
> So we're comparing different indices
> with respect to the same terms; i.e.,
> when a(m) = b(n).  One simple way to
> operationalize "similarity" (or
> "proximity", I suppose) is to find m-n when
> a(m) = b(n).  (There are other
> ways to conceptualize "similarity"; but the key
> here is to use sequences
> which are permutations of each other and evaluate a(m)
> = b(n)).
>
> As an illustration, let A064413(m) be a(m) and let  B255582(n) be
> b(n).  For
> example, consider just these terms:
>
> a(5247) = 5458 =
> b(5250)
> a(5248) = 2729 = b(5252)
> a(5249) = 8187 = b(5254)
>
> m-n = -3, -4, -5
> which is relatively quite small, so m~n; thus the sequences
> are "similar" (at
> least with respect to these three terms; as it turns out,
> it appears to be the
> case for both entire sequences).
>
> However, if we consider a(m) - b(n) where m=n,
> the sequences do not look so
> similar:
>
> a(5247) = 5458, b(5247) = 5457
> a(5248) =
> 2729, b(5248) = 5456
> a(5249) = 8187, b(5249) = 5463
>
> So while a(m) - b(n) m=n
> may be quite large, m-n when a(m) = b(n) is quite
> small.
>
> The general principle
> of comparing sequences with different indices and same
> terms can apply to any
> number of sequences which are permutations of each
> other.   They can be
> evaluated with regard to "similarity" or any number of
> other constructs.
>
> So,
> is there a standard name or way to describe this general principle??  Is
> there
> a name for the specific type of similarity" where  a(m) = b(n),
> m~n??
>
> Cheers,
> Bob
>
>
> --------------------------------------------------
> From:
> "Olivier Gerard" <olivier.gerard at gmail.com>
> Sent: Wednesday, March 04, 2015
> 10:02 PM
> To: "Sequence Fanatics Discussion list"
> <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Evaluating sequences by same terms
> and
> differentindices
>
>> On Wed, Mar 4, 2015 at 10:55 PM, Bob Selcoe
> <rselcoe at entouchonline.net>
>> wrote:
>>
>>>
>>> Hi Seqfans,
>>>
>>> Given two (or
> more) sequences, each with unique terms and where all are
>>> permutations of the
> other, I want to evaluate and perform operations on
>>> the
>>> different indices
> where the terms across sequences are exactly the same.
>>>
>>> The sequences may
> be finite or infinite.
>>>
>>> For example, let sequences A and B be permutations
> of the first eight
>>> positive even numbers:
>>>
>>> A =  8   12   14   2   10   6
> 16   4
>>> B = 10   6    14  12   4    8    2   16
>>>
>>> Let terms in A be a(m)
> and terms in B be b(n).
>>>
>>> Order the terms so a(m) = b(n):
>>>
>>> a(4)=2,
> a(8)=4, a(6)=6, a(1)=8, a(5)=10, a(2)=12, a(3)=14, a(7)=16
>>> b(7)=2, b(5)=4,
> b(2)=6, b(6)=8, b(1)=10, b(4)=12, b(3)=14, b(8)=16
>>>
>>>
>> That's the core of
> the idea of permutation. There are many notations
>> but all things considered
> they are equivalent. Historically it has been
>>
>>
>> What you have just pondered
> is the equivalent of
>>
>> 48615237  is the permutation transforming A into 2n
>>
>>
> 75261438  is the permutation transforming B into 2n
>>
>> So if  2n is considered
> the base = the identity permutation
>> these are respectively the inverse of A
> and B
>>
>> 12345678
>> 64371285   is the permutation transforming A in B
>>
>>
> 12345678
>> 56328147  is the permutation transforming B in A
>>
>>
>>
>>> Now we can
> perform operations with m and n where a(m) = b(n), including
>>> creating new
> sequences.  For example, C = m-n:
>>>
>>> C = -3  3  4  -5  4  -2  0  -1
>>>
>>>
>>
> This is one of the several "Eulerian" themes in combinatorics of
>>
> permutations.
>> If you consider all possible arrangements of A and B and count
> 0s, signs,
>> etc.
>> You end up with the Eulerian Numbers and their
> relatives.
>>
>> If you want to find sequences based on this and similar ideas,
> lookup
>> permutation and
>>
>> ascent, descent, inversions, derangements,
> etc.
>>
>>
>>> Alternatively, we could use the actual order of one sequence as a
> base,
>>> and compare the other sequence accordingly:
>>>
>>> a(1)=8, a(2)=12,
> a(3)=14, a(4)=2, a(5)=10, a(6)=6, a(7)=16, a(8)=4
>>> b(6)=8, b(4)=12, b(3)=14,
> b(7)=2, b(1)=10, b(2)=6, b(8)=16, b(5)=4
>>>
>>> C = -5  -2  0  -3  4  4  -1
> 3
>>>
>>>
>> yes but unless your original function from 1..n to whatever values
> or
>> symbols set has peculiar properties
>> it won't bring you much more than
> considering 1..n directly
>>
>>
>>> Of course, we could do various other operations
> and evaluate multiple
>>> sequences simultaneously.  I suppose we could even
> eliminate the
>>> requirement that the terms be unique, and make indexing rules
> for the
>>> repeated terms.
>>>
>>>
>> This brings to mind the many variations on
> permutations in combinatorics :
>> signed permutations, colored permutations,
>>
> bi-permutations, n-permutations, multiset permutations, etc.
>>
>>
>>> Is there a
> standard name for comparing and evaluating indices of multiple
>>> sequences in
> this way?  Are there any OEIS sequences which use this
>>> approach??
>>>
>>>
>> See
> above for starting points.
>>
>>
>>> Best,
>>> Bob Selcoe
>>>
>>>
>>
>>
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