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We are going to discuss about Young Tableux today !
Given a positive integer n, a partition of n is a tuple (p1, p2, ..., pk) such that p1 >= p2 >= ... >= pk and the sum p1+p2+...+pk = n
(There are 7 partitions of 5 which are
5
4+1
3+1+1
3+2
2+1+1+1
2+2+1
1+1+1+1+1)
Given a partition of n, we can draw it 'Ferrer shape' as follows:
Put p1 blocks in row 1
Put p2 blocks in row 2 and so on and then left justify everything
So corresponding to the partition 5+3+3+2+1+1+1 of 16, we have the Ferrer shape:
* * * * *
* * *
* * *
* *
*
*
*
What is a Standard Young Tableux ?
Well, we have to fill some positive integers into these boxes(or stars) in a way such that entries in any given row or column increase left to right and top to bottom respectively
For example, we can fill the above Ferrer's shape as follows:
3 4 7 8 9
5 9 10
6 13 15
9 17
11
19
21
(You can check that every row and column is increasing)
Q) What is the number of ways to fill a Young Tableux of a given shape with numbers only from the set {1,2,3,...,n} with no repetition (the Ferrer shape has n boxes)
Given a Ferrer shape of a partition of n, we can take its 'transpose' to get another partition of n called as the conjugate partition
In our running example of 5+3+3+2+1+1+1, the ferrer shape as discussed before is:
* * * * *
* * *
* * *
* *
*
*
*
Read it row wise to get (5,3,3,2,1,1,1)
Read it column wise to get (7,4,3,1,1)
Q) When are two partitions p=(p1,...,pn) and q=(q1,...,qm) conjugate to each other ? (It might be difficult at this stage to find sufficient conditions so just find necessary conditions)
Q) When is the conjugate of a partition itself ?
Q) Given an m x n rectangle, how many Ferrer shapes fit into it ? (Does this answer remind you of something ? dimension of some important vector space?)
For example, in the 2 x 2 rectangle
* *
* *
we can fit in "(2+2) choose 2" that is 6 Ferrers shapes
1) **
2) **
**
3) **
*
4) *
*
5) *
6) (empty)
did you guys know about the Kaprekar constant 6174 ?
Take any 4 digit number (with at least 2 different digits) ... say, 8955
Write it in ascending order (5589) and descending order (9855)
Subtract the two to get - 4266
Repeat : 6642-2466 = 4176
Repeat: 7641-1467 = 6174 !
(If you continue with 6174 you keep getting 6174. sounds similar to collatz conjecture 🤔)
Questions (increasing order of difficulty):
1) Prove this fact
2) Explore n-digit constants (of course, with proof)
3) Explore different bases of numbers
Give a function which is continuous on all rationals and discontinuous on irrationals
Give a function which is continuous on all irrationals and discontinuous on rationals
( f : R -> R )
