I think it's this way for the 7 partitions:

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Partition 1: 3,3 

Choose the types for the 2 triples in 36C2 ways.

That's 36C2 = 630 ways.

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Partition 2: 3,1,1,1 

Choose the type for the triple in 36C1 ways
Choose the types for the 3 singles in 35C3 ways

That's 36C1×35C3  = 36×6545 = 235620 ways

[Notice here that if you choose the 3 singles first in 36C3 ways,
then the triple in 33C1 ways,

that would be 36C3×33C1 = 7140×33 = 235620 ways, which shows that
it doesn't matter.] 

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Partition 3: 2,1,1,1,1 

Choose the type for the double in 36C1 ways
Choose the types for the 4 singles in 35C4 ways

That's 36C1×35C4 = 36×52360 = 1884960 ways

[Notice here that if you choose the 4 singles first in 36C4 ways,
then choose the triple in 32C1 ways,

that would be 36C4×32C1 = 58905×32 = 1884960 ways, which shows that
it doesn't matter] 

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Partition 4: 1,1,1,1,1,1 

Choose the types for the 6 singles in 36C6 ways

That's 36C6 = 1947792 ways

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Partition 5: 2,2,2 

Choose the types for the 3 doubles in 36C3 ways

That's 36C3 = 7140 ways.

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Partition 6: 2,2,1,1 

Choose the types for the 2 doubles in 36C2 ways
Choose the types for the 2 singles in 34C2 ways

That's 36C2×34C2 = 630×561 = 353430 ways.

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Partition 7: 3,2,1

Choose the type for the triple in 36C1 ways
Choose the type for the double in 35C1 ways
Choose the type for the single in 34C1 ways

That's 36C1×35C1×34C1 = 36×35×33 = 42840 ways.

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630 + 235620 + 1884960 + 1947792 + 7140 + 353430 + 42840 = 4472412 total.

Edwin