[seqfan] Re: Counting ells.

[William Keith]

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
>