Okay, so I’ve been reading a book about the Oulipo literary movement. I became interested in Queneau numbers, a mathematical concept utilized as a creative constraint by some members. It’s a number sequence described by Oulipo founder and mathematician Raymond Queneau, and reads as follows:
>Queneau numbers: numbers n such that the Queneau-Daniel permutation {1, 2, 3, …, n} -> {n, 1, n-1, 2, n-2, 3, …} is of order n.
Here’s my problem: I have no idea what that means. I can do basic algebra and geometry but most mathematics beyond that escapes me. So, can somebody dumb this down for me?
In: Mathematics
It’s been a while since I’ve had my group theory classes, so if anyone spots a mistake, please point it out.
Let’s start with the beginning: a permutation. If you’ve got a set of numbers, for instance all natural numbers between 1 and n inclusive: {1, 2, …, n-1, n}, then you can rearrange them. You could say “take the kth number, and swap it with the lth number”, turning your set into:
{1, 2, … , l, … , k, … , n}
Or you could say “move everything m places to the left”. Or “turn the set around”, or whatever you can think of, really. To make it more concrete, I’ll take 1 to 5. Our first example, let’s switch the numbers on place 2 and 4.
{1, 2, 3, 4, 5} -> {1, 4, 3, 2, 5)
Move everything 2 places to the left:
{1, 2, 3, 4, 5} -> {4, 5, 1, 2, 3}
Turn the set around:
{1, 2, 3, 4, 5} -> {5, 4, 3, 2, 1}
Now, I kind of didn’t tell the entire truth in the beginning. It can be any set, really, not just numbers. Permutations are also linked to rotations of different shapes (eg rotating a square over 90 degrees is the same as taking 4 consecutive numbers, and moving them one to the left/right), and as a result with the symmetry of different shapes. But let’s focus on numbers for now.
Now, we’ve got a set of numbers, and an operation we can do on it. It makes sense that we can do that same operation on that same set again, right? Let’s take a look at our examples again:
{1, 2, 3, 4, 5} -> {1, 4, 3, 2, 5) -> {1, 2, 3, 4, 5}
because we switch the numbers on place 2 and 4; In the first operation, it’s 2 and 4, but in the second it’s 4 and 2. Our second operation:
{1, 2, 3, 4, 5} -> {4, 5, 1, 2, 3} -> {2, 3, 4, 5, 1}
And finally:
{1, 2, 3, 4, 5} -> {5, 4, 3, 2, 1} -> {1, 2, 3, 4, 5}
Now, you’ve noticed that in some cases, you’ll end up with the original set. So now we’ve got an interesting question. How often can you do your operation in a row before we get to the original set again? This is called the order of the permutation. If it takes 2 permutations, then it’s of order 2. If it takes 3, it’s order 3. If it takes n, it’s of order n. So in our previous examples, we had 2 permutations of order 2, and 1 we don’t know the order of yet. I’ll leave it up to you to find it.
So let’s now take a look at these Queneau numbers. We’ve got our set: 1 to n. And our permutation:
{1, 2, 3, …, n} -> {n, 1, n-1, 2, n-2, 3, …}
So the last number goes first, then the first, then the penultimate, then the 2nd and so on. And n is a Queneau number, if the order of this operation is n. Let’s take a look at some examples to see if we can find some.
I’m skipping 1, because it can be debated whether or not it is one, so let’s just jump into 2:
{1, 2} -> {2,1} -> {1, 2}
We did our permutation twice, which means it’s a Q number (I’m just going to name it that from now on). What about 3?
{1, 2, 3} -> {3, 1, 2} -> {2, 3, 1} -> {1, 2, 3}
Another Q number, as it turns out. 4?
{1, 2, 3, 4} -> {4, 1, 3, 2} -> {2, 4, 3, 1) -> {1, 2, 3, 4)
We’ve found our first non- Q number, since only had to do this 3 times instead of 4 (also, from a really basic hunch, I’m guessing based on nothing that all powers of 2 are non-Q numbers, except for 2)
5?
{1, 2, 3, 4, 5) -> {5, 1, 4, 2, 3} -> {3, 5, 2, 1, 4} -> {4, 3, 1, 5, 2} -> {2, 4, 5, 3, 1} -> {1, 2, 3, 4, 5}
Another Q-number (another guess based on nothing is that it’s linked to prime numbers, which may all be Q-numbers)
And so on.
So, to recap. A permutation is just a rule that says how to change up your set; the order of a permutation is how often you have to repeat it in order to get the original set back; and a Queneau number is a number for which the order of the given permutation is that number.
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