[seqfan] Re: partitions in A120452 and A344613

[William Orrick]

Dear Richard,

A slight clarification: in (2) I should have written, "If the row
containing the *bottom-most element of* the shorter odd column has length
k, ..."

I would also add that the description of A120452 didn't make it immediately
clear to me what the sequence is counting. An alternative description would
be "Number of partitions of n with one marked part, the marked part not
equal to 2."

Best,
Will

On Tue, Jun 8, 2021 at 8:24 AM William Orrick <will.orrick at gmail.com> wrote:

> Dear Richard,
>
> The two sequences are equal. A comment to A344613 says that "a(n) is the
> number of odd-length partitions of 2n whose conjugate partition has exactly
> two odd parts." Consider the Ferrers graph of the partition. It must have
> two odd-length columns and, because the length of the partition is odd, one
> of those must be the first column. There are two cases:
>
> (1) the two odd-length columns are of equal length, and hence must be the
> first two columns. In this case the smallest part is 2 and all rows except
> the last come in equal-length pairs,
>
> (2) the two odd-length columns are unequal, and hence the smallest part is
> 1. If the row containing the shorter odd column has length k, the following
> row must have length k-1. (Otherwise there would be two columns with the
> shorter odd length.) All rows except these two and the last, length-1 row
> come in equal-length pairs.
>
> We can now exhibit a bijection between these partitions and the
> partitions of A120452: create an empty partition, P. Identify a pair of
> same-length rows and add a single part of the same length of the form
> oo...o to P. Then delete the pair. Continue until there are no same-length
> pairs. What remains is either a single length-2 row, in which case we add
> the part * to P, or three rows of lengths k, k-1, and 1, in which case we
> add the part oo...o* to P, where o occurs k-1 times. The partition P will
> never contain o* because that would require k=2 in the latter case, which
> would mean two rows of length 1 (the length-(k-1) row and the length-1
> row). This won't occur because pairs of equal-length rows have already been
> elminated.
>
> -Will Orrick-
>
> On Fri, Jun 4, 2021 at 5:05 AM Richard J. Mathar <mathar at mpia-hd.mpg.de>
> wrote:
>
>> Are these two sequences duplicates (with A344613 an additional a(0).):
>> http://oeis.org/?q=id:A344613|id:A120452
>> There is a simple sum formula in A344613 which may equal one of the
>> partitions formulas in A120452 (supposing one can decipher Mma).
>>
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>