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Euler’s Identity Glaisher’s Bijection

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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

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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

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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.

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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)

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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

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Numbers and Colors

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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

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Fun with Ferrers Diagrams The power of pictures

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Conjugation

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11 10 7 6 5 5 5 2 21 191513113

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15131197

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Durfee Square j≤ j

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Durfee Square

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

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

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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

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

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Bijections with things other than partitions

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

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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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Conjugate Pairs Theorem Every complex polynomial function of degree n 1 has exactly n complex zeros, some of which may repeat. 1) A polynomial function.

Conjugate Pairs Theorem Every complex polynomial function of degree n 1 has exactly n complex zeros, some of which may repeat. 1) A polynomial function.

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