Partitions Based On Permutations

[Max Alekseyev]

On Dec 4, 2007 11:34 PM, Max Alekseyev <maxale at gmail.com> wrote:

> > Does every integer greater than some integer
> > (such as, say, 20) have such a representation?

[...]

> It is interesting to look at the number of representations of numbers
> in the interval [A000292(j),A000330(j)] in the form
> 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j),
> where m is a permutation of order j.

[...]

> It turns out that for j>=4, no number in the interval
> [A000292(j),A000330(j)] has zero number of representations.

Actually, this can be easily proved by induction on j.

> > %C A000001 Does every integer greater than some positive integer N have at least one such representation?
>
> trivially has an affirmative answer.

The reason is that A000330(j)>A000292(j+1) for j>=6 and hence the
union of intervals [A000292(j),A000330(j)] for all j>=5 equals
[35,+oo), implying that every integer n>=35 has at least one
representation of the form:
1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where m is a permutation of order j.

Max