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

[Olivier Gerard]

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
>
>