[seqfan] Re: a partition sequence

[Nathaniel Johnston]

>
> I don't know if it's the smallest, but [4^2,3,2^2,1^3] cannot be written in
> this way.


This can be written as:

4 3 2
4
2
1
1
1

Which seems to mee the criteria to me.

All the best,

- Nathaniel Johnston


On Sun, Oct 2, 2011 at 6:13 PM, <franktaw at netscape.net> wrote:

> I don't know if it's the smallest, but [4^2,3,2^2,1^3] cannot be written in
> this way.
>
> Responding to William Keith's response: I think this is an error; instead
> of
>
> 3 3
>
> It should be
> 3
> 3
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: David Newman <davidsnewman at gmail.com>
>
> I'd like someone to check my numbers before I propose this sequence.
>
> Let a(n) be the number of planar partitions whose summands are
> distinct going across rows and have distinct frequencies going down
> columns.
>
> I get a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=9, a(6)=14, a(7)=21 for the
> values that I've computed by hand.
>
> For example the allowed partitions of 6 are:
>
> 6
>
>
> 5 1
>
>
> 4 2
>
>
> 4
> 1
> 1
>
>
> 3 3
>
>
> 3 2 1
>
>
> 3 1
> 1
> 1
>
>
>
> 3
> 1
> 1
> 1
>
>
>
> 2
> 2
> 2
>
>
> 2 1
> 2
> 1
>
>
> 2 1
> 2 1
>
>
> 2 1
> 1
> 1
> 1
>
> 2
> 1
> 1
> 1
> 1
>
> 1
> 1
> 1
> 1
> 1
> 1
>
>
> Question:  Can every ordinary, unrestricted partition be written as a
> planar
> partition of this sort?  If not, what is the smallest n for which there is
> an unrestricted partition which cannot be written as a planar partition of
> this sort?
>
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