[seqfan] Re: A possible new Keyword
Max Alekseyev
maxale at gmail.com
Thu Sep 24 21:26:25 CEST 2015
I'd rather say that it is possible to enumerate partitions this way (not
that it is based on an ordering of the partitions).
Also, this is very different from other ordering of partitions, which
iterate over partitions of a fixed number first.
Regards,
Max
On Thu, Sep 24, 2015 at 2:49 PM, Frank Adams-Watters <franktaw at netscape.net>
wrote:
> Note that while you can do this without thinking of partitions, it is in
> fact based on an ordering of the partitions: A112798.
>
> Franklin T. Adams-Watters
>
> -----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
> already included in OEIS?
> > Has this problem appeared before?
> >
> > Best
> regards,
> > Juan Arias de Reyna
> >
> >
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> >
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> http://list.seqfan.eu/
> >
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