# [seqfan] Re: A possible new Keyword

Thu Sep 24 20:49:41 CEST 2015

```Note that while you can do this without thinking of partitions, it is in fact based on an ordering of the partitions: A112798.

-----Original Message-----
From: Max Alekseyev <maxale at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Thu, Sep 24, 2015 1:04 pm
Subject: [seqfan] Re: A possible new Keyword

Notice that there is also a way to enter coefficients of polynomials in
infinite
number of variables without referring to partitions.
This is what I call prime
factorization order:

For n with the prime factorization n = p1^k1 * p2^k2 *
..., where p1=2,
p2=3, ... are the primes in their natural order (so only a
finite number of
ki are nonzero),
we let a(n) be the coefficient of u[1]^k1 *
u[2]^k2 * ....

E.g., multinomial coefficients C(k1+k2+...; k1, k2, ...)
arranged this way
give http://oeis.org/A008480

Regards,
Max

On Thu, Sep 24,
2015 at 11:47 AM, Juan Arias de Reyna <arias at us.es> wrote:

>
> Frequently we
consider  sequences of  polynomials in an infinite number of
> variables.
>
>
For example the fourth polynomial in a given sequence I find is
>
> >  - u[1]^4
+ 4 u[1]^2 u[2] - 2 u[2]^2 - 4 u[1] u[3] + 4 u[4]
>
> There is a coefficient for
each partition of  n = 4. In this case each
> coefficient is
> related to a
partition of 4
>
> 4 = 1 + 1 + 1 + 1       coef  -1
>   = 1 + 1 + 2
coef   4
>   = 1 + 3               coef  -4
>   = 2 + 2               coef  -2
>
= 4                   coef   4
>
>
> To include the sequence of coefficients we
want to give a canonical order
> to
> all possible partitions.
> I will consider
useful that all partition of the number n precede all
> partitions of n+1.
>
>
But how to order the partitions of a number n?
> Perhaps writing the summands in
increasing order and then by lexicografic
> order  as  I have
> written above
the partitions of 4.
>
> Maybe somebody here in OEIS has a better election.
> I
will like to fix a particular order,
> then in all cases that a sequence of
>
this type is included in OEIS this same order should be used.  Also a new
>
keyword as the
> actual tabf and tabl should be added to the Keywords file, to
indicate
> this type of sequence.
>
>
> Somebody know sequences of this type
> Has this problem appeared before?
>
> Best
regards,
> Juan Arias de Reyna
>
>
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