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  • We're talking about a problem involving putting numbers into boxes and other shapes.

  • And the shape is an arrangement of boxes that looks something like this.

  • And I've got a certain number of boxes in the first row,

  • so I have four boxes in the first row, and two in the second row, and one in the third row.

  • And the rule is that these numbers must be decreasing.

  • And the second rule is that the boxes are left-justified.

  • And that's it! That's the rule. A simple example would be this, here's a square.

  • Three boxes in the first row, three in the second row, and three in the third row.

  • The number of boxes in two rows can be the same.

  • Another example of a shape would be something like this. This is something we call a staircase.

  • Four, and three, and two, and one.

  • Another shape that's kind of interesting, it turns out, for my purposes,

  • is where I have two rows of the same length.

  • This would be the shape (6,6).

  • Why don't we put numbers in them?

  • And so maybe I can even do it with these shapes and show you exactly what we do.

  • So, in this first shape, we have seven boxes altogether, so I take the numbers 1–7.

  • The rules are that the numbers must be increasing along rows and increasing down columns.

  • So, 1-3-6 is increasing, 2-5 is increasing, and so forth. Those are the rules.

  • This is called a Standard Young Tableau. Young is a person. Yes, that's right.

  • While I'm at it, the word "tableau" just simply means "an arrangement of numbers in a shape,"

  • with no rules satisfying them.

  • For example, I can just do an example of a tableau that's not a Standard Young Tableau.

  • I'll just put the 1 here, and the 2 here, and the 3 here, and the 4 here, and the 5 and the 6 and the 7...

  • So, these numbers are not increasing along rows and increasing down columns.

  • So that's just called a tableau.

  • And we're gonna talk today about Standard Young Tableaux.

  • Here, I'll make a Standard Tableau and actually, while I make it,

  • it's sort of important to understand what I'm doing while I'm making it,

  • because I have to put a 1 in the upper-left-hand corner, there's no choice about that.

  • And then 2 has to go either here or here, I'll put it there.

  • 3 has to go either here or here at that point, I'll put it down here.

  • Maybe 4 here, 5 here.

  • And if I stop before I'm finished, you can see that what's happening is that I'm building the shape.

  • The shape is the 3x3 square, but I'm building it up cell by cell.

  • In terms of what I'm getting, it's a sequence of subshapes.

  • So right now, what I've done, I've created a tableau of this shape.

  • But I'll finish it up now, so 4, 5, 6, 7, 8, 9.

  • So that's a Standard Tableau of that shape.

  • I'll do one here quickly. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

  • And here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

  • It's easy to do it because you just sort of build your way from the upper-left-hand corner

  • down to the bottom right hand corner in this case.

  • Are these hard to make? Are there lots of them?

  • Are they rare? Are Standard Young Tableaux common?

  • Well, that's a good question, because what I really want to talk about is how many there are.

  • The problem of computing the number of Standard Tableaux.

  • What's kind of amazing is that we know very exact and precise answers to that question.

  • So, I'll tell you, for example, let's take this shape.

  • The number of Standard Young Tableaux—I'll abbreviate it as SYTis 42.

  • So, I could systematically write down all 42 tableaux,

  • and I could convince myself that was correct,

  • but there's a formula for it, and the formula goes like this, and I'll show you how you do it.

  • It's equal to, let's see, there's nine cells in this shape.

  • So it's 9×8×7×6×5×4×3×2×1, divided bythat's 9!—and by the way,

  • that's the number of ways of just filling this shape with the numbers in any manner whatsoever.

  • So the total number of fillings of this shape

  • which is what we've called a tableau

  • is 9!—it's this very large numberbecause

  • there's 9 choices for what to put here,

  • and 8 for what to put there once you've decided on the first one,

  • and 7 for what you put here once you've decided on these two, etc.

  • So you get 9! quite easily.

  • But if you require the rules to be satisfied,

  • then the number has to be reduced by something,

  • and I'm gonna tell you what exactly it has to be reduced by.

  • First of all, you take this shape, and we fill it up with a bunch of other numbers,

  • and the numbers are constructed this way:

  • I put a 1 here, and then a 2, and then a 3, and then I go up, 4, and then 5,

  • and then I go down, 4 and then 3, and then down, 2, and then up, 3,

  • and I divide this by the product of all those numbers.

  • And the ratio of these two numbers is the number of Standard Young Tableaux of this shape.

  • Now let's just compute it. So these cancel...

  • I've canceled everything and I see 3×2×7, which is 42.

  • So that's how I got 42.

  • Now I guess the million-dollar question is what on earth are these numbers?

  • That's the question, the big question! Well, there are two questions.

  • One, where did these numbers come from? They just came out of the blue!

  • What's going on? And secondly, why is it true? And why is it true generally?

  • So what's gonna happen is that not only have I shown you a formula for computing

  • Standard Tableaux of shapes like this, but it's gonna work for any shape.

  • So I have to tell you what these numbers are.

  • These numbers are called the hook numbers.

  • Hook numbers are the following: you have a shape—I'll do it sort of schematically first.

  • Every cell has a hook number.

  • You can see, for example, here. Every cell had a hook number.

  • How did I get this hook number?

  • The answer is that I first of all computed the hook associated with that cell.

  • And the hook is everything that you get by going down from it,

  • and to the right, and the square itself.

  • So you get this sort of inverted-L-shaped thing, which we call a hook.

  • So that's the hook associated with that cell.

  • And so, for example, we can take this cell here. What's its hook number?

  • Well, its hook consists of everything to the right and everything below,

  • and so there are three cells altogether, so the hook number is 3.

  • This has hook number 4, because I have one, two, three, four.

  • This has hook number 5, so I have one, two, three, four, five, etc.

  • So that's what I was doing when I went through that crazy process of filling the square up with numbers.

  • I was computing the hook numbers.

  • Actually, so the main result now is that for any shapethe kinds of shapes I was telling

  • you about before, not just these special square shapesthe formula is exactly this.

  • You take n!, where n is the number of cells, and divide it by the product of all the hook numbers.

  • All right, so let's do one more example.

  • Here's the shape. That's a (4,2,1) shape.

  • Yes, exactly. Well first of all, n is 7.

  • So the number of Standard Young Tableaux is going to be

  • 7×6×5×4×3×2×1 divided by the product of the hook lengths.

  • So, what are they? I'm going to start putting the hook lengths in each cell.

  • This is what we call a corner cell.

  • Maybe I should use that term now, because it's going to come up again.

  • It's a cell with hook length 1. Its hook number is 1.

  • And that's a corner cell, and that's also a corner cell.

  • This one has hook number 2,

  • because the hook only goes to the right. There's two cells in it.

  • This has hook number 4.

  • This has hook number 3.

  • This has hook numberwhat is it?—6.

  • One, two, three, four, five, six.

  • And so the formula says that I should take 7! and divide by 6×4×2×1×3×1×1.

  • So let's do that computation. Cancel the 3's...

  • Cancel the 4's, cancel the 2's, cancel the 6's.

  • And it looks like we get 7×5, or 35.

  • The number of Standard Young Tableaux of this shape is 35.

  • —I'm always surprised by how high these numbers are for such simple shapes.

  • You asked a good question before. If you fix the number of cells, which ones are the biggest?

  • Which ones give you the largest number?

  • That's an interesting and rather difficult question

  • that people do know the answer to,

  • but it's too complicated to talk about today.

  • This video was filmed at the Mathematical Sciences Research Institute,

  • in that room there actually, but the interview didn't stop here.

  • If you'd like to go deeper into the world of tableaux and hook numbers,

  • including some details about how to prove all this,

  • the interview continues over at our second channel, Numberphile2.

We're talking about a problem involving putting numbers into boxes and other shapes.

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形狀和鉤形數字 - 數字愛好者 (Shapes and Hook Numbers - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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