[seqfan] triangular partitions

[David Newman]

I'd like someone to please check the numbers that I get for these
partitions.  It has been my experience that I almost always make some
mistakes in my calculations by hand.

The idea is to count partitions in two dimensions, not into a square array,
as is done in "plane partitions", but into a triangular array.  I mean a
triangle, as in "Pascal's triangle". (I do not mean that the numbers are
triangular numbers)

I picture a triangle with one entry on the first row, two entries on the
second, etc.  Each entry is an integer.  The entries in each row, each
diagonal, and each antidiagonal are non-increasing.

The numbers which I get for this scheme are 1,1,2,4,7,12,21,34,56 which
matches A003293 <http://oeis.org/A003293>. ( I will make a wild guess that
these two sequences are identical, but I have no good reason to think so.)

For example  the partitions of 4 are :

4


   3
1    0


   2
2     0


    2
1     1


          2
     1        0
1        0        0


          1
      1       1
   1      0     0

and


                1
            1      0
        1      0      0
    1       0     0      0








A second set of rules gives numbers not in the OEIS.   Here is the second
set of rules.  Diagonals and anti-diagonals are strictly decreasing, rows
are non-increasing.  For this I get the sequence 1,1,2,3,4,6,9,12,17,22.

For example the partitions of 4 with these rules are:

4


   3
1    0


   2
1     1


I haven't as yet tried computing sequences for other sets of rules such as:
Diagonals, anti-diagonals, and rows are all strictly decreasing.