[seqfan] Re: Counting ells.

William Keith william.keith at gmail.com
Mon May 23 07:48:53 CEST 2022

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.)

William Keith

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