“Fluvius, planus et altus, in quo et agnus ambulet et elephas natet,” wrote Pope Gregory the Great (540-604). “There’s a river, wide and deep, where a lamb may wade and an elephant swim.” He was talking about the Word of God, but you can easily apply his words to mathematics. However, in the river of mathematics, the very shallow and the very deep are often a single step apart.

Here’s a good example. Take the integer 2. How many different ways can it be represented as an sum of separate integers? Easy. First of all it can be represented as itself: 2 = 2. Next, it can be represented as 2 = 1 + 1. And that’s it. There are two partitions of 2, as mathematicians say:

2 = 2 = 1+1 (p=2)

Now try 3, 4, 5, 6:

3 = 3 = 1+2 = 1+1+1 (p=3)
4 = 4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1 (p=5)
5 = 5 = 1+4 = 2+3 = 1+1+3 = 1+2+2 = 1+1+1+2 = 1+1+1+1+1 (p=7)
6 = 6 = 1+5 = 2+4 = 3+3 = 1+1+4 = 1+2+3 = 2+2+2 = 1+1+1+3 = 1+1+2+2 = 1+1+1+1+2 = 1+1+1+1+1+1 (p=11)

So the partitions of 2, 3, 4, 5, 6 are 2, 3, 5, 7, 11. That’s interesting — the partition-counts are the prime numbers in sequence. So you might conjecture that p(7) = 13 and p(8) = 17. Alas, you’d be wrong. Here are the partitions of n = 1..10:

1 = 1 (p=1)
2 = 2 = 1+1 (p=2)
3 = 3 = 1+2 = 1+1+1 (p=3)
4 = 4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1 (p=5)
5 = 5 = 1+4 = 2+3 = 1+1+3 = 1+2+2 = 1+1+1+2 = 1+1+1+1+1 (p=7)
6 = 6 = 1+5 = 2+4 = 3+3 = 1+1+4 = 1+2+3 = 2+2+2 = 1+1+1+3 = 1+1+2+2 = 1+1+1+1+2 = 1+1+1+1+1+1 (p=11)
7 = 7 = 1+6 = 2+5 = 3+4 = 1+1+5 = 1+2+4 = 1+3+3 = 2+2+3 = 1+1+1+4 = 1+1+2+3 = 1+2+2+2 = 1+1+1+1+3 = 1+1+1+2+2 = 1+1+1+1+1+2 = 1+1+1+1+1+1+1 (p=15)
8 = 8 = 1+7 = 2+6 = 3+5 = 4+4 = 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4 = 2+3+3 = 1+1+1+5 = 1+1+2+4 = 1+1+3+3 = 1+2+2+3 = 2+2+2+2 = 1+1+1+1+4 = 1+1+1+2+3 = 1+1+2+2+2 = 1+1+1+1+1+3 = 1+1+1+1+2+2 = 1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1 (p=22)
9 = 9 = 1+8 = 2+7 = 3+6 = 4+5 = 1+1+7 = 1+2+6 = 1+3+5 = 1+4+4 = 2+2+5 = 2+3+4 = 3+3+3 = 1+1+1+6 = 1+1+2+5 = 1+1+3+4 = 1+2+2+4 = 1+2+3+3 = 2+2+2+3 = 1+1+1+1+5 = 1+1+1+2+4 = 1+1+1+3+3 = 1+1+2+2+3 = 1+2+2+2+2 = 1+1+1+1+1+4 = 1+1+1+1+2+3 = 1+1+1+2+2+2 = 1+1+1+1+1+1+3 = 1+1+1+1+1+2+2 = 1+1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1+1 (p=30)
10 = 10 = 1+9 = 2+8 = 3+7 = 4+6 = 5+5 = 1+1+8 = 1+2+7 = 1+3+6 = 1+4+5 = 2+2+6 = 2+3+5 = 2+4+4 = 3+3+4 = 1+1+1+7 = 1+1+2+6 = 1+1+3+5 = 1+1+4+4 = 1+2+2+5 = 1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3 = 1+1+1+1+6 = 1+1+1+2+5 = 1+1+1+3+4 = 1+1+2+2+4 = 1+1+2+3+3 = 1+2+2+2+3 = 2+2+2+2+2 = 1+1+1+1+1+5 = 1+1+1+1+2+4 = 1+1+1+1+3+3 = 1+1+1+2+2+3 = 1+1+2+2+2+2 = 1+1+1+1+1+1+4 = 1+1+1+1+1+2+3 = 1+1+1+1+2+2+2 = 1+1+1+1+1+1+1+3 = 1+1+1+1+1+1+2+2 = 1+1+1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1+1+1 (p=42)

It’s very simple to understand what a partition is, but very difficult to say how many partitions, p(n), a particular number will have. Here’s a partition: 11 = 4 + 3 + 2 + 2. But what is p(11)? Is there a formula for the sequence of p(n)?

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 3118 5, 37338, 44583, 53174, 63261... (A000041 at the OEIS)

Yes, there is a formula, but it is very difficult to understand the Partition function that supplies it. So that part of the river of mathematics is very deep. But a step away the river of mathematics is very shallow. Here’s another question: If you multiply the numbers in a partition of n, what’s the largest possible product? Try using the partitions of 5:

4 = 1 * 4
6 = 2 * 3
3 = 1 * 1 * 3
4 = 1 * 2 * 2
2 = 1 * 1 * 1 * 2
1 = 1 * 1 * 1 * 1 * 1

The largest product is 6 = 2 * 3. So the answer is easy for n = 5, but I assumed that as n got bigger, the largest product got more interesting, using a subtler and subtler mix of prime factors. I was wrong. You don’t have to struggle to find a formula for what you might call the maximum multiplicity of the partitions of n:

1 = 1 (n=1)
2 = 2 (n=2)
3 = 3 (n=3)
4 = 2 * 2 (n=4)
6 = 2 * 3 (n=5)
9 = 3 * 3 (n=6)
12 = 2 * 2 * 3 (n=7)
18 = 2 * 3 * 3 (n=8)
27 = 3 * 3 * 3 (n=9)
36 = 2 * 2 * 3 * 3 (n=10)
54 = 2 * 3 * 3 * 3 (n=11)
81 = 3 * 3 * 3 * 3 (n=12)
108 = 2 * 2 * 3 * 3 * 3 (n=13)
162 = 2 * 3 * 3 * 3 * 3 162(n=14)
243 = 3 * 3 * 3 * 3 * 3 (n=15)
324 = 2 * 2 * 3 * 3 * 3 * 3 (n=16)
486 = 2 * 3 * 3 * 3 * 3 * 3 (n=17)
729 = 3 * 3 * 3 * 3 * 3 * 3 (n=18)

It’s easy to see why the greatest prime factor is always 3. If you use 5 or 7 as a factor, the product can always be beaten by splitting the 5 into 2*3 or the 7 into 2*2*3:

15 = 3 * 5 < 18 = 3 * 2*3 (n=8)
14 = 2 * 7 < 24 = 2 * 2*2*3 (n=9)
35 = 5 * 7 < 72 = 2*3 * 2*2*3 (n=12)

And if you’re using 7 → 2*2*3 as factors, you can convert them to 1*3*3, then add the 1 to another factor to make a bigger product still:

14 = 2 * 7 < 24 = 2 * 2*2*3 < 27 = 3 * 3 * 3 (n=9)
35 = 5 * 7 < 72 = 2*3 * 2*2*3 < 81 = 3 * 3 * 3 * 3 (n=12)

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Post-Performative Post-Scriptum

The title of this post is, of course, a paronomasia on core Beatles album Live and Let Die (1954). But what does it mean? Well, if you think of the partitions of n as slivers of n, then you sliv n to find its partitions:

9 = 9 = 1+8 = 2+7 = 3+6 = 4+5 = 1+1+7 = 1+2+6 = 1+3+5 = 1+4+4 = 2+2+5 = 2+3+4 = 3+3+3 = 1+1+1+6 = 1+1+2+5 = 1+1+3+4 = 1+2+2+4 = 1+2+3+3 = 2+2+2+3 = 1+1+1+1+5 = 1+1+1+2+4 = 1+1+1+3+3 = 1+1+2+2+3 = 1+2+2+2+2 = 1+1+1+1+1+4 = 1+1+1+1+2+3 = 1+1+1+2+2+2 = 1+1+1+1+1+1+3 = 1+1+1+1+1+2+2 = 1+1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1+1 (p=30)

And when you find the greatest product among those partitions, you let 3 or “tri” work its multiplicative magic. So you “Sliv and Let Tri”:

8 = 1 * 8
14 = 2 * 7
18 = 3 * 6
20 = 4 * 5
7 = 1 * 1 * 7
12 = 1 * 2 * 6
15 = 1 * 3 * 5
16 = 1 * 4 * 4
20 = 2 * 2 * 5
24 = 2 * 3 * 4
27 = 3 * 3 * 3 ←
6 = 1 * 1 * 1 * 6
10 = 1 * 1 * 2 * 5
12 = 1 * 1 * 3 * 4
16 = 1 * 2 * 2 * 4
12 = 1 * 2 * 3 * 3
24 = 2 * 2 * 2 * 3
5 = 1 * 1 * 1 * 1 * 5
8 = 1 * 1 * 1 * 2 * 4
9 = 1 * 1 * 1 * 3 * 3
12 = 1 * 1 * 2 * 2 * 3
16 = 1 * 2 * 2 * 2 * 2
4 = 1 * 1 * 1 * 1 * 1 * 4
6 = 1 * 1 * 1 * 1 * 2 * 3
8 = 1 * 1 * 1 * 2 * 2 * 2
3 = 1 * 1 * 1 * 1 * 1 * 1 * 3
4 = 1 * 1 * 1 * 1 * 1 * 2 * 2
2 = 1 * 1 * 1 * 1 * 1 * 1 * 1 * 2
1 = 1 * 1 * 1 * 1 * 1 * 1 * 1 * 1 * 1