[seqfan] Re: a partition sequence

[franktaw at netscape.net]

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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