+0

Combinatorics

+12
139
11
+85

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.)

Jul 26, 2022

#1
0

There are 8 partitions that work.

Jul 26, 2022
#2
+118132
+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

Jul 26, 2022
#4
+85
+12

Thank you for your work! It helped me to arrive at a final answer of 12.

Doremy  Jul 27, 2022
#5
+118132
+5

I didn't get 12

What 12 did you get?

Melody  Jul 27, 2022
#6
+85
+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
+118132
+4

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
+85
+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
#9
+118132
+2

I asked you to list them Doremy.

Why have you not done so.

Melody  Jul 29, 2022
#10
+85
+7

5421

5331

5322

53211

4431

4422

44211

4332

43311

43221

6321

432111

Total—12 partitions that work.

Doremy  Jul 30, 2022
#11
+118132
+1

Thanks Doremy.

Melody  Jul 31, 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.

Jul 26, 2022
edited by Guest  Jul 26, 2022