A partition of a positive integer n means a way of writing as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of 4: $$4\qquad 3+1\qquad 2+2\qquad 2+1+1\qquad 1+1+1+1$$

How many partitions of 12 are there that have at least four parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to 4,3,2,1?

(The partition 12 = 4 + 4 + 2 +2 is one such partition.)

Doremy Jul 26, 2022

#2**+7 **

4+3+2+1=10 so only 2 more can be added

1,2,3, 6

1,2,4, 5

1,3,3, 5

2,2,3, 5

1,2, 5,4

1,3, 4,4

2,2, 4,4

1, 4,3,4

2, 3,3,4

3,2,3,4

That is all of them but some are copies. You can count how many are originals

Melody Jul 26, 2022

#6**+11 **

It should be C5,2 but there are 2 additional cases for when both are added on the same one. We can also use complementary counting and get 16 - 4(the repeated ones)

Doremy
Jul 28, 2022

#7**+4 **

Thank your for resonding.

12 is a small number, I challenge you to list them.

I can se no relevance at all of 5C2. Perhaps you would like to explain?

Melody
Jul 28, 2022

#8**+8 **

4 3 2 1, is the original partition of 10. However, we can have a hidden "0" at the end, making it 4,3,2,1,0 for our use in calculating the partitions of 12.

For 12, we can place two extra 1's into the five existing slots (4 3 2 1 0), in other words five choose two places. Note that five choose two = 10.

At this point the answer seems like 10 but there are two additional cases not counted, which are to place both extra 1's into the same place AND to create 2 new slots at the end instead of one. There are two ways to do so, leaving us with respectively 6,3,2,1 and 4,3,2,1,1,1

So the answer is 10+2 = 12. I was originally kinda stuck but figured it out later.

Idk if this is faster than just listing everything out but its probably more elegant

Doremy
Jul 29, 2022

#3**+1 **

There are a total of 15 partitions with 4 parts as follows:

9 + 1 + 1 + 1 = 12

8 + 2 + 1 + 1 = 12

7 + 3 + 1 + 1 = 12

7 + 2 + 2 + 1 = 12

6 + 4 + 1 + 1 = 12

6 + 3 + 2 + 1 = 12

6 + 2 + 2 + 2 = 12

5 + 5 + 1 + 1 = 12

5 + 4 + 2 + 1 = 12

5 + 3 + 3 + 1 = 12

5 + 3 + 2 + 2 = 12

4 + 4 + 3 + 1 = 12

4 + 4 + 2 + 2 = 12

4 + 3 + 3 + 2 = 12

3 + 3 + 3 + 3 = 12

Note: Simply pick the ones that meet your conditions.

Guest Jul 26, 2022

edited by
Guest
Jul 26, 2022