Euler’s Identity

1. Euler’s Identity Glaisher’s Bijection

2. Let l be a partition of n into odd parts If you have two i-parts i+i in the partition, merge them to form a 2i-part. Continue merging pairs until no pairs remain

3. Let m be a partition of n into distinct parts Split each even part 2i into i+i Repeat this splitting process until only odd parts are left

4. A Generalization Glaisher’s Theorem: The same splitting/merging process can be used, except you merge d-tuples in one direction and split up multiples of d in the other.

5. Another Generalization: Euler Pairs Definition: A pair of sets (M,N) is an Euler pair if Theorem (Andrews): The sets M and N form an Euler pair iff (no element of N is a multiple of two times another element of N, and M contains all elements of N along with all their multiples by powers of two)

6. Examples of Euler Pairs Euler’s Identity Uniqueness of binary representation

7. Numbers and Colors

8. Scarlet Numbers 1 +1 +1 +1 +1 +1 +1 +1 +1+1+1 +1+1+1 +1 +1 1 +1+1+1+1 +1 ø 1+1+1+1+1 +1

9. Scarlet Numbers 1

10. Fun with Ferrers Diagrams The power of pictures

11. Conjugation

12. 21 19 15 13 11 3 11 11 10 10 10 7 6 5 5 5 2

13. 15 13 11 9 7 11 11 11 11 11

14. j ≤ j ≤ j Durfee Square

15. Durfee Square

16. A Beautiful Bijection By Bressoud 17 15 12 8 2 Indent the rows

17. 16 Odd rows on top (decreasing order) 12 6 Even rows on bottom (decreasing order) 11 9 A Beautiful Bijection By Bressoud

18. A Boxing Bijection By Baxter Definition: For positive integers m, k, an m-modular k-partition of n is a partition such that: • There are exactly k parts • The parts are congruent to one another modulo m

19. 31 26 21 16 11 6 6 6 6 6 25 10 5 6 6 25 15 A Boxing Bijection By Baxter

20. Bijections with things other than partitions

21. Plane Partitions Weakly decreasing to the right and down

22. The number of plane partitions which fit in an n×n×n cube The number of tilings of a regular hexagon by diamonds