Laura, the author of MathSux2, offered this week’s A-to-Z term. (I apologize for it being late but the Playful Math Education Blog Carnival 141 work took a lot out of me.) She writes the blog weekly, and hosts a YouTube channel of mathematics videos also. I’m glad to have the topic to discuss.
We learn to count permutations before we know what they are. There are good reasons to. Counting permutations gives us numbers that are big, and therefore interesting, fast. Counting is easy to motivate. Humans like counting. Counting is useful. Many probability questions are best answered by counting all the ways to arrange things, and how many of those arrangements are desirable somehow.
The count of permutations asks how many ways there are to put some things in order. If some of the things are identical, the number is smaller. Calculating the count may be a little tedious, but it’s not hard. We calculate, rather than “really” count, because — well, list all the possible ways to arrange the letters of the word ‘DEMONSTRATION’. I bet you turn that listing over to a computer too. But what is the computer counting?
If we’re trying to do this efficiently we have some system. Start with ‘DEMONSTRATION’. Then, say, swap the last two letters: ‘DEMONSTRATINO’. Then, mm, move the ‘N’ to the antepenultimate position: ‘DEMONSTRATNIO’. Then, oh, swap the last two letters again: ‘DEMONSTRATNOI’.
Then, oh, move the ‘N’ to the third-to-the-last position: ‘DEMONSTRANTIO’. What next? Oh, swap the last two letters again: ‘DEMONSTRANTOI’. Or, move what been the last letter to the antepenultimate position: ‘DEMONSTRANOTI’. And swap the last two letters once more: ‘DEMONSTRANOIT’.
Enough of that, you and my spellchecker say. I agree. What is it that all this is doing? What does that tell us about what a permutation is?
An obvious thing. Each new variation of the order came from swapping two letters of an earlier one. We needed a sequence of swaps to get to ‘DEMONSTRANOIT’. But each swap was of only two things. It’s a good thing to observe.
Another obvious thing. There’s no letters in ‘DEMONSTRANOIT’ or any of the other variations that weren’t in ‘DEMONSTRATION’. All that’s changed is the order.
This all has listed eight permutations, counting the original ‘DEMONSTRATION’ as one. There are, calculations tell me, 778,377,592 to go.
Would the number of permutations be different if we were shuffling around different things? If instead of the letters in the word ‘DEMONSTRATION’ it were, say, the numerals in the sequence ‘1234567897045’? Or the sequence of symbols ‘!@#$%^&*(&)$%’ instead? No, and that it would not is another clue about what permutations are.
Another thing, obvious in retrospect. Grant that we’ve been making new permutations by taking a sequence of letters (numerals, symbols) and swapping a pair. We got from ‘DEMONSTRATION’ to ‘DEMONSTRATINO’ by swapping the last two letters. What happens if we swap the last two letters again? We get ‘DEMONSTRATION’, a sequence of letters all right, although one already on our list of permutations.
One more thing, obvious once you’ve seen it. Imagine we had not started with ‘DEMONSTRATION’ but instead ‘DEMONSTRATNIO’. But that we followed the same sequences of swappings. Would we have come up with different permutations? … At least for the first couple permutations? Or would they be the same permutations, listed in a different order?
You’ve been kind, letting me call these things “permutations” before I say what a permutation is. It’s relied on a casual, intuitive idea of a permutation. It’s a shuffling around of some set of things. This is the casual idea that mathematicians rely on for a permutation. Sure we can make the idea precise. How hard will that be?
It’s not hard in form. The permutation is the rearranging of things into a new order. The hard part is the concept. It’s not “these symbols in this order” that’s the permutation. It’s the act of putting them in this new order that is. So it’s “swap the 12th and the 13th symbols”. Or, “move the 13th symbol to 11th place, the 11th symbol to 12th, and the 12th symbol to 13th place”.
We can describe each permutation as a function. All the permutation functions have the same domain and the same range. And the range is the domain. The function is a bijection. Every item in the domain matches exactly one item in the range, and vice-versa. There’s some sequence for the elements in the domain. And the rule for the function describes how that sequence changes.
So one permutation is “swap the 12th and the 13th elements”. Another permutation is “swap the 11th and the 12th elements”. Since the range of one function is the domain of another, we can compose the together. That is, we can “swap the 12th and the 13th elements, and then swap the 11th and the 12th elements”. This gets us another permutation. The effect of these two permutations, in this order, is “make the 13th element the 11th, make the 11th element the 12th, and make the 12th element the 13th”. The order we do these permutations in counts. “Swap the 11th and the 12th elements, and then swap the 12th and the 13th” gets us a different net effect. That one is “make the 12th element the 11th, make the 13th element the 12th, and make the 11th element the 13th”. Composition of functions does not commute.
That functions compose is normal enough. That their composition doesn’t commute is normal enough too. These functions are a bit odd in that we don’t care what the domain-and-range is. We only care that we can index the elements in it. That leads us to some new observations.
The big one is that the set of all these permutations is a group. I mean the way mathematicians mean group. That is, we have a set of items. These are the functions, the permutations. The instructions, like, “make the 12th element the 11th and the 13th element the 12th”, or “the 12th element the 13th”. We also need a group action, a thing that works like addition does for real numbers. That’s composition. That is, doing one permutation and then the other, to get a new permutation out of it. That new permutation is itself one of the permutations we’d had. We can’t compose permutations and get something that’s not a permutation. No amount of swapping around the letters of ‘DEMONSTRATION’ will get us ‘DEMONSTRATIONERS’.
When we talk about how permutations as a group work, we want to give individual permutations names. That ends up being letters. These are often Greek letters. I don’t know why we can’t use the ordinary Latin alphabet. I suppose someone who liked Greek letters wrote a really good textbook and everyone copies that. So instead of speaking about x and y, we’ll get α and β. Sometimes σ and τ. Or, quite often π, especially if we need a bunch of permutations. Then we get π1, π2, π3, and so on. πj. All the way to πN. For the young mathematics major it might be the first time seeing π used for something not at all circle-related. It’s a weird sensation. Still, αβ is the composition of permutation α with permutation β. This means, do permutation β first, and then permutation α on whatever that result is. This is the same way that f(g(x)) means “evaluate g(x) first, and then figure out what f( that ) is”.
That’s all fine for naming them. But we would also like a good way to describe what a permutation does. There are several good forms. They all rely on indexing the elements, using the counting numbers: 1, 2, 3, 4, and so on. The notation I’ll share is called cycle notation. It’s easy to type. You write it within nice ordinary parentheses: (11 12) means “put the 11th element in slot 12, and the 12th element in slot 11”. (11, 12, 13) means “put the 11th element in slot 12, the 12th element in slot 13, and the 13th element in slot 11”. You can even chain these together: (10, 11)(12, 13) means “put the 10th element in slot 11 and the 11th element in slot 10; also, put the 12th element in slot 13, and the 13th element in slot 12”.
In that notation, writing (9), for example, means “put the 9th element in slot 9”. Or if you prefer, “leave element 9 alone”. Or we don’t mention it at all. The convention is that if something isn’t mentioned, leave it where it is.
This by the way is where we get the identity element. The permutation (1)(2)(3)(4)(etc) doesn’t actually swap anything. It counts as a permutation. Doing this is the equivalent of adding zero to a number.
This cycle notation makes it not hard to figure out the composition of permutations. What does (1 2)(1 3) do? Well, the (1 3) swaps the first and the third items. The (1 2), next, swaps what’s become the first and the second items. The effect is the same as the permutation (2 3 1). You can get pretty good at this sort of manipulation, in time.
You may also consider: if (1 2)(1 3) is the same as (2 3 1), then isn’t (2 3 1) the same as (1 2)(1 3)? Sure. But, like, can we write a longer permutation, like, (1 3 5 2 4), as the product of some smaller permutations? And we can. If it’s convenient, we can write it as a string of swaps, exchanging pairs of elements. This was the first “obvious” thing I had listed. A long enough chain of pairwise swaps will, in time, swap everything.
We call the group made of all these permutations the Symmetric Group of the set. Since it doesn’t matter what the underlying set is, just the number of elements in it, we can abbreviate this with the number of elements. S2. S4. SN. Symmetric Groups are among the first groups you meet in abstract algebra that aren’t, like, integers modulo 12 or symmetries of a triangle. It’s novel enough to be interesting and to not be completely sure you’re doing it right.
You never leave the Symmetric Group, though, not if you stay in algebra. It has powerful consequences. It ties, for example, into the roots of polynomials. The structure of S5 tells us there must exist fifth-degree polynomials we can’t solve by ordinary arithmetic and root operations. That is, there’s no version of the quadratic equation for high-order polynomials, and never can be.
There are more groups to build from permutations. The next one that you meet in Intro to Abstract Algebra is the Alternating Group. It’s made of only the even permutations. Those are the permutations made from an even number of swaps. (There are also odd permutations, which are what you imagine. They can’t make a group, though. No identity element.) They’re great for recapturing dread and uncertainty once you think you’ve got a handle on the Symmetric Group.
They lead to other groups too, and even rings. The Levi-Civita symbol describes whether a set of indices gives an even or an odd permutation (or neither). It makes life easier when we work on determinants and tensors and Jacobians. These tie in to the geometry of space, and how that affects physics. It also gets a supporting role in cross products. There are many cryptography schemes that have permutations at their core.
So this is a bit of what permutations are, and what they can get us.
Today’s and all the other 2020 A-to-Z essays should be at this link. Both the All-2020 and past A-to-Z essays should be at this link. Thanks for reading.