[seqfan] Sequence idea

[jnthn stdhr]

Hi all,

A sequence idea that is *omino-like.  I'm doing this by hand, so forgive me
if I've made an error and this already exists.

How many unique constructions, not counting rotations or reflections, are
formed using n rectangles?

For n > 0, take the list of partition sets from row n in A080577 (lists all
partitions of n, in graded reverse lexicographic order) and create blocks
of rectangles that correspond to the values in each set. Rectangles are
glued together along long edges to form blocks. Now count how many unique
ways we can stack each set of blocks.  A construction is valid as long as
blocks share at least one edge, which allows overhanging.

As an example of overhanging, one partition of 5 is (2,2,1) and one of the
five constructions based on this partition is:  a block of two rectangles
sits atop a single rectangle that sits atop a block of two.

We can produce an irregular triangle that has row lengths = P(n), row
elements list the number of constructions for each partition of n, and row
sums are the total number of constructions for n.  The triangle begins:

1                                 = 1
1 1                              = 2
1 1 1                           = 3
1 2 3 2 1                      = 9
1 2 2 6 5 3 1                = 20
1 3 3 10 3 16 4 4 8 5 1 = 58

Using rows four and five, I get no match for the triangle.  Row sums almost
match A029895 and A073268.

If no mistakes were made, does this look interesting?  If so, I will
allocate two A-numbers and post them in a follow-up.

Jonathan