Complete partitions

[Augustine Munagi]

Hi,
This seems like the newest meaning to be associated with  "complete".

For instance

complete composition (A107429) => set of parts forms and interval of
consecutive integers including 1.
(gapfree composition (A107428) => 1 is not required).

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}.

(see also "r-complete partition", Discrete Math. 183 (1998) 293–297;
"M-partition", Discrete Math. 306 (2006) 694-698).

Cheers!

Augustine


On 3/23/07, franktaw at netscape.net <franktaw at netscape.net> wrote:
> I just sent in the following comment:
>
> %I A126796
> %S A126796
> 1,1,2,2,4,5,8,10,16,20,31,39,55,71,100,125,173,218,291,366,483,600,784,97
> 1,1244,
> 1538,1957,2395,3023,3693,4605,5604,6942,8397,10347,12471,15235,18309,2226
> 7,26619,
> 32219,38414,46216,54941,65838,77958,93076,109908,130615,153855,182248,213
> 961,
> 252631,295913,348145,406826,477288,556230,650852,756881
> %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.
> %o A126796 (PARI)
> T(n,k)=if(k<=1,1,if(n<2*k-1,T(n,floor((n+1)/2)),T(n,k-1)+T(n-k,k)))
> a(n)=T(n,floor((n+1)/2)) /* If modified to save earlier results, this
> would be efficient. */
> %O A126796 1
> %K A126796 ,nonn,
> %A A126796 Franklin T. Adams-Watters (FrankTAW at Netscape.net), Mar 22
> 2007
>
> It seems to me that, based on this, it should be possible to find a
> generating function for this sequence.  I don't see how to, however.
> Can somebody else on this list do it?
>
> (The T(n,k) in the PARI program are the number of complete partitions
> of n whose largest part is <= k.  Note the similarity to the triangle
> of partition numbers, A008284.)
>
> Franklin T. Adams-Watters
>
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