non-triangular tables and their complement

[Wouter Meeussen]

hello,

recently, I twice encountered a situation where I got a 
"non-triangular table" like the following:

{1}
{0, 1}
{0, 1, 1}
{0, 0, 1, 1}
{0, 0, 1, 1, 1}
{0, 0, 1, 1, 1, 1}
{0, 0, 0, 2, 1, 1, 1}
{0, 0, 0, 1, 2, 1, 1, 1}
{0, 0, 0, 1, 2, 2, 1, 1, 1}
{0, 0, 0, 1, 2, 2, 2, 1, 1, 1}
{0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1}
{0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1}

with a "complement" like 

{1}
{1, 0}
{1, 1, 0}
{1, 1, 0, 0}
{1, 2, 0, 0, 0}
{1, 2, 1, 0, 0, 0}
{1, 3, 1, 0, 0, 0, 0}
{1, 3, 2, 0, 0, 0, 0, 0}
{1, 4, 3, 0, 0, 0, 0, 0, 0}
{1, 4, 4, 1, 0, 0, 0, 0, 0, 0}
{1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0}
{1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0}

What's special? The abundance of zero's makes it a bit
artificial & unelegant to describe each of these collections of
numbers as a 'triangular table'.

But, where one got a zero, the other has a non-zero entry & vice versa.
For rows with "special" index, there is just a tiny overlap with value "1".

-----------------------------
Instance One (illustrated above):
Count the Q-partitions of n with first element m;
        complement:
Count the Q-partitions of n with m elements;
special index : n=k(k+1)/2
-----------------------------
Instance Two (not illustrated):
Count the partitions of n whose Ferrers-plot just covers a m*m square;
        complement:
Count the partitions of n whose Ferrers-plot is just covered by a m*m square;
special index: n=k^2
-----------------------------

Question:
        Such instances must be quite common. It comes down to
mutually exclusive conditions (with a tiny overlap). 
How many of these are already in EIS? In what format?
How many were never submited because of their awkward 'un-triangular' format?

wouter.

PS. I recently constructed a Mathematica few-liner that produces the list of
symmetric (=self-transpose) partitions of 128 in about 32 seconds.
Just in case anyone would have use for it.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be