Example from worksheet continued

Example 2: the partitions with a Durfee square of size three with odd parts.

A partition with a Durfee square of size 3 with only odd parts can be decomposed into two pieces (lambda, mu) where lambda is an odd partition with parts of size less than or equal to 3 consisting of the 4th and later parts, and mu is the partition consisting of the first three parts. The generating function for this set of partitions will be the product of the generating functions for the odd partitions of size less than or equal to 3 times the generating function for the odd partitions of length equal to 3 containing a 3x3 square.

The generating function for the odd partitions with parts less than or equal to 3 is the generating function for partitions using only parts of size 1 or 3 and so is equal to 1/(1-q)/(1-q^3).

Now a partition with 3 odd parts which contains a 3x3 Durfee square has 3+2*a_1 columns of size 3, 2*a_2 columns of size 2 and 2*a_3 columns of size 1. The generating function for the columns of size three is
(q^3)^3/(1-(q^3)^2) = q^9 + q^15 + q^21 + q^27 + ...
Similarly, the generating function for the columns of size 2 and 1 are 1/(1-(q^2)^2) and 1/(1-q^2) respectively. Therefore the generating function for the partitions with three odd parts containing a 3x3 Durfee square is q^9/(1-q^6)/(1-q^4)/(1-q^2).

The generating function for the odd partitions containing a Durfee square of size three is the product of the two generating functions we found above so it is
q^9/(1-q^6)/(1-q^4)/(1-q^3)/(1-q^2)/(1-q)

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