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N 1 2 3 4 5 6 7 8 9 An 42 429 7436 218348 10850216 911835460 = 2  3  7 = 3  11  13 = 22  11  132 = 22  132  17  19 = 23  13  172  192 = 22.

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Presentation on theme: "N 1 2 3 4 5 6 7 8 9 An 42 429 7436 218348 10850216 911835460 = 2  3  7 = 3  11  13 = 22  11  132 = 22  132  17  19 = 23  13  172  192 = 22."— Presentation transcript:

1 n 1 2 3 4 5 6 7 8 9 An 42 429 7436 218348 = 2  3  7 = 3  11  13 = 22  11  132 = 22  132  17  19 = 23  13  172  192 = 22  5  172  193  23 1 –1

2 n 1 2 3 4 5 6 7 8 9 An 42 429 7436 218348 very suspicious 1 –1 = 2  3  7 = 3  11  13 = 22  11  132 = 22  132  17  19 = 23  13  172  192 = 22  5  172  193  23

3 There is exactly one 1 in the first row
2 3 4 5 6 7 8 9 An 42 429 7436 218348 There is exactly one 1 in the first row 1 –1

4 There is exactly one 1 in the first row
2 3 4 5 6 7 8 9 An 1+1 2+3+2 There is exactly one 1 in the first row 1 –1

5 1 1 –1

6 There is exactly one 1 in the first row
2 3 4 5 6 7 8 9 An 1+1 2+3+2 1 –1

7 There is exactly one 1 in the first row
2 3 4 5 6 7 8 9 An 1+1 2+3+2 1 –1

8 1 1 –1 + + +

9 1 1 –1 + + +

10 1 1 2/2 1 2 2/ /2 2 7 2/ /2 7 42 2/ /2 42 429 2/ /2 429 1 –1

11 1 1 2/2 1 2 2/ /2 2 7 2/ / /2 7 42 2/ / / /2 42 429 2/ / / / /2 429 1 –1

12 2/2 2/ /2 2/ / /2 2/ / / /2 2/ / / / /2 1 –1

13 1+1 1+2 1+1 2+3 1+3 1+1 3+4 3+6 1+4 1+1 4+5 6+10 4+10 1+5 Numerators:
1 –1

14 Numerators: 1+1 1+1 1+2 1+1 2+3 1+3 1+1 3+4 3+6 1+4
Numerators: 1 –1 Conjecture 1:

15 Conjecture 2 (corollary of Conjecture 1):
–1 Conjecture 2 (corollary of Conjecture 1):

16 1 –1 Richard Stanley M.I.T.

17 George Andrews, Penn State
1 –1 Richard Stanley M.I.T. Andrews’ Theorem: the number of descending plane partitions of size n is George Andrews, Penn State

18 All you have to do is find a 1-to-1 correspondence between n by n alternating sign matrices and descending plane partitions of size n, and conjecture 2 will be proven! 1 –1

19 What is a descending plane partition?
All you have to do is find a 1-to-1 correspondence between n by n alternating sign matrices and descending plane partitions of size n, and conjecture 2 will be proven! 1 –1 What is a descending plane partition?


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