[seqfan] Re: Counting ells.

[Allan Wechsler]

William Keith's aim is as true as William Tell's. My attempt to construct a
partition model was incoherent; William did it right, and A270060 is indeed
the sequence I was looking for. Thank you, William!

(Now I would like to see about 9,925 more terms. I might have to write some
code.)

On Mon, May 23, 2022 at 1:57 AM William Keith <william.keith at gmail.com>
wrote:

> Note that by the way you worded this in the last condition 331 should also
> qualify, since its largest part is not smaller than its number of parts,
> but it is the conjugate of 322 and I believe your intent is that these
> reflections should not be considered distinct.
>
> Best,
> William
>
> On Mon, May 23, 2022 at 1:48 AM William Keith <william.keith at gmail.com>
> wrote:
>
> > By restricting yourself to partitions which have largest part no smaller
> > than the number of parts, you are casting out half of the
> > non-self-conjugate partitions.  One may take the sequence for partitions
> > into two part sizes (A002133), add the sequence for the number of
> > representations of n as a difference of two squares (A100073), which
> counts
> > the self-conjugate partitions into two part sizes, and halve the
> result.  I
> > get A270060, "Number of incomplete rectangles of area n," which seems to
> be
> > your object.
> >
> > Your 5 should apparently be a 6 but your listed terms are otherwise
> > correct.  (I calculate that 7 has relevant partitions 61, 52, 511, 43,
> > 4111, 322.)
> >
> > Best,
> > William Keith
> >
>
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