Shapes and Hook Numbers - Numberphile
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Date: 2022-04-08
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Comments and reviews: 9
piguy314159
Not being an expert in combinatorics, I did some programming and found the shapes that give the most possible tableaux for small n:
1 cell: 1 (1)
2 cells: 2 (1)
3 cells: 2-1 (2)
4 cells: 3-1 (3)
5 cells: 3-1-1 (6)
6 cells: 3-2-1 (16)
7 cells: 4-2-1 (35)
8 cells: 4-2-1-1 (90)
9 cells: 4-3-1-1 (216)
10 cells: 4-3-2-1 (768)
)
55 cells: 12-10-8-6-5-4-3-2-2-1-1-1 (1. 12 - 10-35)
So it seems like staircases are the winners at first, but the winner for 55 looks more like the region between a square and its inscribed circle. Maybe that's the magic family?
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Not being an expert in combinatorics, I did some programming and found the shapes that give the most possible tableaux for small n:
1 cell: 1 (1)
2 cells: 2 (1)
3 cells: 2-1 (2)
4 cells: 3-1 (3)
5 cells: 3-1-1 (6)
6 cells: 3-2-1 (16)
7 cells: 4-2-1 (35)
8 cells: 4-2-1-1 (90)
9 cells: 4-3-1-1 (216)
10 cells: 4-3-2-1 (768)
)
55 cells: 12-10-8-6-5-4-3-2-2-1-1-1 (1. 12 - 10-35)
So it seems like staircases are the winners at first, but the winner for 55 looks more like the region between a square and its inscribed circle. Maybe that's the magic family?
reply
theo7
The question at 4: 20 seems retty easy to answer once you have established the hook number formula. The minimum combinations of a given number of cells is given by maximum hook number sum (which we take if the tableau is just a line and in fact there is only 1 possible combination) and the maximun number of combinations is given by the minimum hook number sum which we take if we stack the squares on diagonals 1 by 1. The interesting question is to prove the hook number formula.
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The question at 4: 20 seems retty easy to answer once you have established the hook number formula. The minimum combinations of a given number of cells is given by maximum hook number sum (which we take if the tableau is just a line and in fact there is only 1 possible combination) and the maximun number of combinations is given by the minimum hook number sum which we take if we stack the squares on diagonals 1 by 1. The interesting question is to prove the hook number formula.
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Jesus
Now you need to work out how many squares you would need to construct a square of squares, by only allowing squares within the shapes with matching values to touch the squares in bordering shapes
Like a mathematical tetris
The questions are:
Could you ever construct a square of squares at all, by following this rule?
Or not?
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Now you need to work out how many squares you would need to construct a square of squares, by only allowing squares within the shapes with matching values to touch the squares in bordering shapes
Like a mathematical tetris
The questions are:
Could you ever construct a square of squares at all, by following this rule?
Or not?
reply
Sujoy
If we have a square matrix with side length n=1000, how to calculate the hook numbers? We cannot do manually in that case. There should be some formula for hook numbers too. Any square 's hook number is x+y+1 if it is in (n-x) th row and in (n-y) th column where (1
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If we have a square matrix with side length n=1000, how to calculate the hook numbers? We cannot do manually in that case. There should be some formula for hook numbers too. Any square 's hook number is x+y+1 if it is in (n-x) th row and in (n-y) th column where (1
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The
I need a general formula for the Kostka numbers making up the transition matrix from elementary symmetric functions, e, to monomial symmetric functions, m. Following the notation of Ian McDonald's -Symmetric Functions & Hall Polynomials- book.
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I need a general formula for the Kostka numbers making up the transition matrix from elementary symmetric functions, e, to monomial symmetric functions, m. Following the notation of Ian McDonald's -Symmetric Functions & Hall Polynomials- book.
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BunniBuu
That's difficult to talk about? Wouldn't it just be the largest shape with the smallest number of cumulative hooks? I'd guess it would be a stair pattern excluding corners would be the most efficient in that case
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That's difficult to talk about? Wouldn't it just be the largest shape with the smallest number of cumulative hooks? I'd guess it would be a stair pattern excluding corners would be the most efficient in that case
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Stiggandr1
Hey Numberphile. I know you likely will not see this, but I thought I'd suggest that you do an episode on the mathematics behind go. I imagine that if you like the math behind chess you'll love this.
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Hey Numberphile. I know you likely will not see this, but I thought I'd suggest that you do an episode on the mathematics behind go. I imagine that if you like the math behind chess you'll love this.
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Thomas
So the hook number describes the number of cells (including itself) that is must be bigger than; all the cells below it must be smaller, as must all the cells to the right.
I see how this works.
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So the hook number describes the number of cells (including itself) that is must be bigger than; all the cells below it must be smaller, as must all the cells to the right.
I see how this works.
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Juhani
It would've been nice if he had first explained what he's even doing. ?
he's saying there's a problem with putting numbers in boxes and then he just starts putting numbers in boxes.
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It would've been nice if he had first explained what he's even doing. ?
he's saying there's a problem with putting numbers in boxes and then he just starts putting numbers in boxes.
reply
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