[seqfan] Re: Seeking proof of a claim in the comments for A000009

[Jack Grahl]

I think there is a bijection between 'gapless' partitions and distinct
partitions which is just given by reflecting the partition diagram.

Eg
4,3,2,2,1, a partition without gaps
****
***
**
**
*

maps to
5,4,2,1, a distinct partition
*****
****
**
*

On Sun, 26 Sep 2021, 14:34 Allan Wechsler, <acwacw at gmail.com> wrote:

> In the comments for https://oeis.org/A000009, Jon Perry claims:
>
> "Number of partitions of n where if k is the largest part, all parts 1..k
> are present."
>
> I wrote some code and verified that this is true, and it's *plausible*, but
> does anybody have a proof? It would suffice to put these "gapless"
> partitions into correspondence with either partitions into odd parts, or
> partitions with distinct parts.
>
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