I will introduce my question with the following example:

For the partition of 26 = 1+1+1+2+3+3+5+5+5 let us calculate its "fancy number" as follows:

From the group of the ones, we get: 1*(1+1)*(1+1+1)=6
From the group of the twos (actually only one), we get: 2
From the group of the thress, we get: 3*(3+3)=18
From the group of the fives, we get: 5*(5+5)*(5+5+5)=750

That is, for each distinct part we calculate the product of the cumulative sums obtained with that part.

Then, the fancy number of the partition 1+1+1+2+3+3+5+5+5 is the product of the above numbers: 6*2*18*750 = 162000

Another example: for the partitions of 6, the fancy numbers (fn) are:

for 1+1+1+1+1+1 fn is 1*2*3*4*5*6=720 for 1+1+1+1+2 fn is 1*2*3*4*2=48 for 1+1+1+3 fn is 1*2*3*3=18 for 1+1+2+2 fn is 1*2*2*4=16 for 1+1+4 fn is 1*2*4=8 for 1+2+3 fn is 1*2*3=6 for 2+2+2 fn is 2*4*6=48 for 2+4 fn is 2*4=8 for 3+3 fn is 3*6=18 for 1+5 fn is 1*5=5 for 6 fn is 6

which shows that two partitions can have the same fancy number. Now, my question is:

Do the "fancy numbers" recieve any special name? Have they been studied? What is it known about them?

thank you very much!