[seqfan] Re: Given any partition of n into k parts, what's the maximum permutations of parts

[William Keith]

On Fri, Apr 12, 2019 at 8:11 AM Rob Pratt <robert.william.pratt at gmail.com>
wrote:

> If I understand the problem correctly, an optimal partition for n = 512, k
> = 64 is as follows (in frequency representation form):
> 1^7 2^6 3^6 4^5 5^4 6^4 7^4 8^3 9^3 10^3 11^2 12^2 13^2 14^2 15^2 16^2
> 17^1 18^1 19^1 20^1 21^1 22^1 23^1
>

Mr. Pratt:

     I agree that this looks like a pretty good candidate.  I do not see
any local moves that would increase the number of permutations.  Do you
have a proof that this is optimal?  I would be persuadable that it is
always the case the frequencies must increase, by at most 1 at a time, as
part sizes decrease by 1 at a time, which heavily restricts the set of
possible optimal partitions, but I would like to see the argument for that,
and what hits optimality?

Cordially,
William Keith