Partitioning Integers

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Partitioning Integers

• Sharon Jancha
• Dr. Lisa Rome, Mentor

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Math Problem

• Total 7 crayons
• Some are red
• Some are blue
• How many of each could you have?

43
25
61
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Math Problem

• Total 7 crayons
• How many of each color if
• Three colors?
• Four colors?
• Seven colors?
• How many ways can we have 7 crayons using at most
  7 colors?

322
3211
1 of each color
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S. Ramanujan

• From India
• Born Dec 22, 1887
• Died Apr 26, 1920
• Mostly self taught
• 30 publications since his death

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Ramanujan and Hardy

• 1913 1919
• Partitions
• Approximating formula

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Definition

• Partition function, p(n), is the number of ways
  to write the integer n as a sum of non-negative
  integers less than or equal to n.

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Examples
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Example p(5)

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

p(5) 7
221
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Visual RepresentationsRectangular Blocks

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

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• Leading Part is a1

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Example p(5)

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

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Partitions sub m

• The number of partitions of an integer n with no
  part bigger than m.
• Written as pm(n)

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Example p3(5)

• 5
• 41
• 32
• 311
• 221
• 2111
• 11111

p3(5) 5
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Differences in pm(5)
1

1 2 2 1 1 7 p(5)

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Differences in pm(4)
1 2 1 1 5 p(4)
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Differences in pm(3)
1 1 1 3 p(3)
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Differences in pm(2)
1 1 2 p(2)
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Differences in pm(1)
1 p(1)
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Triangle
p(5) 7
1 2 2 1 1 7
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Triangle
p3(5) 5
1 2 2 1 1 5
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• R row number
• E element number in row

R 4 E 3
1
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Build 7th Row

• 2 simple steps
• Which previous row
• Sum the correct elements

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Row 7 1st element

• Find R E
• 7 - 1 6
• Add the first E elements
• E 1

1
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Why 1?

• Element Number Leading part

E 1
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Row 7 2nd element

• R 7 and E 2
• R E 7 2 5
• Add first E terms

3
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Why 3?

• Element Number Leading part

E 2

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Row 7 3rd element

• 7 3 4
• Add first 3 terms

4
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Row 7 4th element

• 7 4 3
• Add first 4
• terms

3
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Row 7 5th element

• 7 5 2
• Add first 5
• terms

2
30
Row 7 6th element

• 7 6 1
• Add first 6
• terms

1
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Row 7 7th element

• 7 7 0

1
32
Why 1?

• Element Number Leading part

E 7
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p(7) ?
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p(7)
How many ways can 7 crayons be divided into at
most 7 colors?
15
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Further areas of study

• Drawn p(1) through p(20)
• Distinct verses non-distinct partitions
• Odd and Even partitions
• Extended triangle
• Proving patterns within

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Thank you to

• My family and friends
• Dr. Rome, and Prof. Dunlap
• Faculty, and Staff

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Questions?