Complete partitions

[Augustine Munagi]

Ok with zero parts, your comment is fine.
But you need to mention the need to reckon with zero parts in the comment.
It is not obvious. The original sequence does not require zero parts to stand.


On 3/23/07, franktaw at netscape.net <franktaw at netscape.net> wrote:
> No, [2,3,3] doesn't fulfill the condition, because that condition
> applies to all the parts, even the smallest.  The parts smaller than 2
> sum to 0, and 2 > 0 + 1.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: aomunagi at gmail.com
>
>  Oh! I see your point.
>  But your condition is not sufficient. 2+3+3 fulfills your condition,
>  but not complete since it does not contain the partition of 1. Also
>  2+2+4, etc.
>
>  On 3/23/07, franktaw at netscape.net <franktaw at netscape.net> wrote:
>  > This is (essentially) the definition used by this sequence. Viz, the
>  > existing comment in the sequence: " A partition of n is complete if
>  > every number 1 to n can be represented as a sum of parts of the
>  > partition." My comment is to show an alternative condition for
>  > completeness (in this sense), not an alternative definition.
>  >
>  > Franklin T. Adams-Watters
>  >
>  > -----Original Message-----
>  > From: aomunagi at gmail.com
>  >
>  > Hi,
>  > This seems like the newest meaning to be associated with "complete".
>  >
>  > For instance
>  >
>  > ...
>  >
>  > Complete partition (Fibonacci Quart. 36 (1998) 354–360) => a
> partition
>  > of n is complete if each smaller integer can be written as a linear
>  > combination of its parts with coefficients in {0,1}.
>  >
>  > ...
>  >
>  > Augustine
>  >
>  > On 3/23/07, franktaw at netscape.net <franktaw at netscape.net> wrote:
>  > > I just sent in the following comment:
>  > >
>  > ...
>  > > %N A126796 Number of complete partitions of n.
>  > > %C A126796 A partition is complete iff each part is no more than 1
>  > more
>  > > than the sum of all smaller parts.
>
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-- 
Augustine O. Munagi
The John Knopfmacher Centre for Applicable Analysis and Number Theory
School of Mathematics
University of the Witwatersrand
Johannesburg 2050
South Africa
Phone: 27-11-717-6246
Fax: 27-11-717-6259