[seqfan] Re: A conjecture of George Beck about A008484

[Rick Shepherd]

Hello all,

George Beck's original conjecture says, in other words, that A008484 counts
the partitions of n where 2/3 of the parts have size 1.

His observation and Martin Fuller's proof can be generalized in the
following way:

Let k, s, d be positive integers with k >= 3, s < k/2, and d  <= ceil(k/s)
- 2. A bijection exists from the set of partitions of n into parts of size
at least k to the set of partitions of n where d/(d+1) of the parts have
size s and 1/(d+1) of the parts have size at least k - ds >= s + 1 (note
well that equality may not be the case in the last relation here). Thus for
a given pair of such k and n, the numbers of such partitions in each set
are equal.

The general proof follows since each original part of size p >= k  can
itself be (sub)partitioned into p - ds, s, s, ..., s, with d occurrences of
s. (It's also easy to see here how the d/(d+1) ratio occurs.)

A logical place to insert this (or a better) generalization (subject to
final checking and correction) would be the sequence (if it exists yet in
the OEIS; I've barely looked) corresponding to A008484 but for which k = 3
(not 4, like A008484 itself, where also s = 1 and d = 2).

Also, while I'm at it -- unless I'm unaware of some (strange to me)
convention:  If someone modifies A008484, the name of the sequence should
clarify that it only applies for n > 0 and that a(0) = 1. (I'm guessing
that the initial 1 comes from the use of a generating function.)

Best regards,
Rick

On Fri, Aug 18, 2023 at 11:46 PM Allan Wechsler <acwacw at gmail.com> wrote:

> That certainly looks sound to me. Can you add it to the entry?
>
> On Fri, Aug 18, 2023 at 9:18 AM Martin Fuller via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
> > 3 * (number of ones) = 2 * (number of parts)
> > <=> 3 * (number of ones) = 2 * ((number of ones) + (number of parts >=
> 2))
> > <=> Number of ones = 2 * (number of parts >= 2)
> >
> > Remove all the ones and increase each other part by 2.  This gives a
> > partition with the same total in which all parts >= 4.
> > Vice versa, decrease each part >= 4 by 2, and add 2 ones for each reduced
> > part.
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
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