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

[Allan Wechsler]

I reworded the proof provided by Jack and Frank (both came up with
essentially the same demonstration), and, crediting both of them, added it
to the comments right after Jon Perry's statement. If I was too terse
somebody should expand it. I used a couple of $1.50 words for brevity.

On Tue, Sep 28, 2021 at 2:54 PM Frank Adams-watters via SeqFan <
seqfan at list.seqfan.eu> wrote:

> Looking at the Ferrers diagram, it is obvious that there are columns of
> the same height iff there are parts of the partition that are equal; this
> follows directly from the definitions.
>
> Let q(k) be the number of parts of a partition Q that are greater than or
> equal to k. Then if k is a missing value k in the parts, q(k) must equal
> q(k-1); and thus there are rows of the Ferrers diagram that have the same
> number of parts; and conversely.
>
> The columns of Q become rows of its conjugate.
>
> So we have Q is composed of distinct parts iff its diagram has all
> distinct columns iff the diagram of Conj(Q) has all distinct rows iff
> Conj(Q) is a stairstep partition. Q.E.D.
>
>
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Allan Wechsler <acwacw at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Sat, Sep 25, 2021 9:29 pm
> Subject: [seqfan] Seeking proof of a claim in the comments for A000009
>
> 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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