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Euler’s Identity Glaisher’s Bijection. Let 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.

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Presentation on theme: "Euler’s Identity Glaisher’s Bijection. Let 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."— Presentation transcript:

1 Euler’s Identity Glaisher’s Bijection

2 Let 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  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 NM {1,3,5,7,9,...}{1,2,3,4,5,6,...} {1}{1,2,4,8,...} Euler’s Identity Uniqueness of binary representation

7 Numbers and Colors

8 Scarlet Numbers ø

9 1

10 Fun with Ferrers Diagrams The power of pictures

11 Conjugation

12

13

14 Durfee Square j≤ j

15 Durfee Square

16 A Beautiful Bijection By Bressoud Indent the rows

17 A Beautiful Bijection By Bressoud Odd rows on top (decreasing order) Even rows on bottom (decreasing order)

18 A Boxing Bijection By Baxter Definition: For positive integers m, k, an m- modular k-partition of n is a partition such that: 1.There are exactly k parts 2.The parts are congruent to one another modulo m

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


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