1) If you want to see the generating functions for r-partitions with exactly N parts and the first part exactly M, here are some samples:
2) If you want to see the guessed polynomials for the coefficient of the ith term from the last of the generating functions with the first part fixed and N ranging from 5 to 20, here are some samples:
3) If you want to see the guessed polynomials (proven using recurrence relation) for the generating functions with the number of parts N fixed (ranging from 1 to 5), here are some samples:
1) If you want to see the generating functions of (r1, r2)-plane partitions whose base is an M by N rectangle and whose first entry is equal to K, here are some samples:
1) If you want to see the generating functions of (r1, r2)-plane partitions whose base is an M by N rectangle and whose first entry is equal to K (with weight q^(sum of parts)/product of parts), here are some samples:
A picture of the fractional count can be seen here. The x, y axis is the first column [k1,k2] of the plane partition and the z (vertical) axis is the fractional counting of plane partitions of 120 (with weight q^(sum of parts)/product of parts).
The commands used to generate this picture are:
with(plots):
listplot3d([seq([seq(Bnk2(120,k1,k2),k1=1..120)],k2=1..120)], color = "LightBlue");
