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

[Frank Adams-watters]

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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