euler’s identity glaisher’s bijection. let be a partition of n into odd parts if you have two...
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Euler’s Identity
Glaisher’s Bijection
)|()|( partsdistinctnppartsoddnp
1 112
)1()1(
1
n n
nn
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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
),5,3,1( 531 mmm 11111
122
14
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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
14
122
11111
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A Generalization
Glaisher’s Theorem:
)|(
)|(
,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.
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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)
)|()|( MinpartsdistinctnpNinpartsnp
MMNMM 2 and 2
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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
)}6(mod1|{ mm )}3(mod1|{ mm
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Numbers and Colors
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Scarlet Numbers
}),1,2,3,...{in parts|()(0
npkpn
k
1
5
23
14
122
113
1112
11111
4
22
13
112
1111
3
12
111
2
11
+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
1ø
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Scarlet Numbers
}),1,2,3,...{in parts|()(0
npkpn
k
1
1
1 1 )1()1(
1
1
1
n
nn
n
rr
qqq
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Fun with Ferrers Diagrams
The power of pictures
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Conjugation)parts|()partlargest |( knpknp )parts|()parts all|( knpknp
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)partsodddistinct|()conjugateself|( npnp 11
11
10
10
10
7
6
5
5
5
2
21 19 15 13 11 3
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)rectangle long|()2by differ parts econsecutiv|( npnp
11
11
11
11
11
15 13 11 9 7
} divides |{)2by differ parts econsecutiv|( ndndnp
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Durfee Square
j m
jmjnpjmpnp parts) |()parts |()( 2
j {≤ j{ ≤ j{
m
jpjpjnp parts) |()parts |()side Durfee|( m mjn 2
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Durfee Square
j m
jmjnpjmpnp parts) |()parts |()( 2
m
jpjpjnp parts) |()parts |()side Durfee|( m mjn 2
12222
1 )1()1()1(1
12
nn
n
nn qqq
q
q
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A Beautiful Bijection By Bressoud
parts)) odd(#2parteven each parts,distinct |(
2)by differ parts all|(
np
np
17151282
Indent the rows
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A Beautiful Bijection By Bressoud
parts)) odd(#2parteven each parts,distinct |(
2)by differ parts all|(
np
np
16126119
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
}))1(,,2,,{in parts|(
partition)-modular -|(
mkmmkknp
kmnp
31262116116
6 6 6 6 6 6 25 25 15 10 5
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Bijections with things other than partitions
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Plane Partitions
Weakly decreasing to the right and down
3 2 2 1
3 1 1
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