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

[Neil Sloane]

Certainly put the proof right after the claim.

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Tue, Sep 28, 2021 at 1:20 PM Allan Wechsler <acwacw at gmail.com> wrote:

> 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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> >
>
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