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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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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. Continue merging pairs until no pairs remain

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

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.

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)

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

Numbers and Colors

Scarlet Numbers 1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1+1+1 +1+1+1+1 +1+1+1+1 +1+1+1+1+1+1 +1+1+1+1+1+1 +1+1+1+1+1+1+1+1 1+1+1+1+11+1+1+1+1 1 ø

1

Fun with Ferrers Diagrams The power of pictures

Conjugation

11 10 7 6 5 5 5 2 21 191513113

15131197

Durfee Square j≤ j

Durfee Square

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

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

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

A Boxing Bijection By Baxter 31 26 21 16 11 6 6 6 6 6 6 625 15 105

Bijections with things other than partitions

Plane Partitions Weakly decreasing to the right and down 3221 311 2 1

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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