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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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122
14
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:
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,2any For
timesdrepeatedpartnonp
dbydivisblepartnonp
d
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)
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MMNMM 2 and 2
Examples of Euler Pairs
N M
{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
Scarlet Numbers
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1
5
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122
113
1112
11111
4
22
13
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1111
3
12
111
2
11
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+1
+1
+1
+1
+1+1
+1+1
+1+1
+1+1+1
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1+1+1+1+1
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Scarlet Numbers
}),1,2,3,...{in parts|()(0
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k
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Fun with Ferrers Diagrams
The power of pictures
Conjugation)parts|()partlargest |( knpknp )parts|()parts all|( knpknp
)partsodddistinct|()conjugateself|( npnp 11
11
10
10
10
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6
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21 19 15 13 11 3
)rectangle long|()2by differ parts econsecutiv|( npnp
11
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15 13 11 9 7
} divides |{)2by differ parts econsecutiv|( ndndnp
Durfee Square
j m
jmjnpjmpnp parts) |()parts |()( 2
j {≤ j{ ≤ j{
m
jpjpjnp parts) |()parts |()side Durfee|( m mjn 2
Durfee Square
j m
jmjnpjmpnp parts) |()parts |()( 2
m
jpjpjnp parts) |()parts |()side Durfee|( m mjn 2
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A Beautiful Bijection By Bressoud
parts)) odd(#2parteven each parts,distinct |(
2)by differ parts all|(
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Indent the rows
A Beautiful Bijection By Bressoud
parts)) odd(#2parteven each parts,distinct |(
2)by differ parts all|(
np
np
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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
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partition)-modular -|(
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Bijections with things other than partitions
Plane Partitions
Weakly decreasing to the right and down
3 2 2 1
3 1 1
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
