[seqfan] Re: Evaluating sequences by same terms and differentindices

[Bob Selcoe]

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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