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

[Allan Wechsler]

I just got around to looking at this. I'm convinced.

Jack, maybe you could add this as a comment? It would make for smoother
reading if the proof were next to the claim; on the other hand comments are
mostly in chronological order. I don't know if OEIS has an editorial policy
about that.

On Sun, Sep 26, 2021 at 10:43 AM Jack Grahl <jack.grahl at gmail.com> wrote:

> 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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> >
>
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