# Find p ( 11 ), p ( 12 ) where pp ( n ) denotes the number of partitions of n

2
by afseri

2015-05-18T11:11:19+05:30

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The partitions of 11 are:

The answer seems to be the sum of   1, 1, 2, 3, 5, 7, 11, 11, 10,  5, 1  =  57    and in standard websites, the p(11) is given as 56.

11
10 + 1
9 + 1 + 1      or  9 + 2
8 + 1 + 1 + 1    or  8 + 1 + 2   or 8 + 3
7 + 1 + 1 + 1 + 1  or  7 + 2 + 1 + 1  or  7 + 2 + 2  or  7 + 3 + 1  or  7 + 4

6+1+1+1+1+1, 6+2+1+1+1, 6+2+2+1, 6+3+2, 6+3+1+1,6+4+1, 6+5

5+1+1+1+1+1+1,  5+2+1+1+1+1, 5+2+2+1+1, 5+2+2+2, 5+3+1+1+1,
5+3+2+1, 5 +3+3, 5+4+1+1, 5+4+2, 5+5+1, 5+6

4+1+1+1+1+1+1+1, 4+2+1+1+1+1+1, 4+2+2+1+1+1, 4+2+2+2+1,
4+3+1+1+1+1, 4+3+2+1+1, 4+3+2+2, 4+3+3+1, 4+4+1+1+1, 4+4+2+1,
4+4+3,           we haven’t  4+5+2, 4+5+1+1, 4+6+1, 4+7

3+1+1+1+1+1+1+1, 3+2+1+1+1+1+1+1, 3+2+2+1+1+1+1, 3+2+2+2+1+1,
3+2+2+2+2,  3+3+1+1+1+1+1, 3+3+2+1+1+1, 3+3+2+2+1, 3+3+3+2, 3+3+3+1+1,
we haven’t 4+4+3, 4+3+1+1+1+1, 4+3+2+1+1, 4+3+2+2, 4+3+3+1, 5+3+1+2,
5 +3+3, 5+3+1+1+1, , 6+3+2, 6+3+1+1, 7+3+1, 8+3

2+2+2+2+2+1,2+2+2+2+1+1+1, 2+2+2+1+1+1+1+1, 2+2+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1,

1+1+1+1+1+1+1+1+1+1+1

p(11) = 56 or 57   by enumerating all possibilities.
==========================

By partition theorem:
g(i) = (3 i² - i) / 2   = Euler's pentagonal number

for all i  positive and negative except 0.

g(-4) = 25,   g(-3) = 15,  g(-2) = 7,  g(-1) = 2 ,  g(1) = 1,  g(2) = 5,  g(3) = 12,  g(4) = 22

Partition(11) =  ...- Part(11-25) + Part(11-15) - Part(11-7) + Part(11 -2) + part(11-1) - Part(11-5) + part(11-12) - Part(11-22) + ....
Partition (11)  =  0+...+0 + 0 - Part(4) + Part(9) +Part(10) - Part(6) + 0 - 0 +...

p(11) = p(9) + p(10) - p(4) - p(6) = 56

p(10) = p(9) + p(8) - p(3) - p(5) = 42
p(9) = p(8) + p(7) - p(4) - p(2) = 30
p(8) = p(7)+p(6)- p(3)- p(1) = 22
p(7) = p(6)+p(5) - p(2) - p(0) =  15
p(6) = p(5)+p(4) - p(1) = 11
p(5) = p(4+p(3) - p(0) = 7
p(4) = p(3)+p(2) = 5      as P(4) = {1+1+1+1, 1+2+1, 2+1, 3+1 ,4}
p(3) = p(2)+p(1) = 3      as P(3) = { 1+1+1 , 1+2 , 3}
p(2) = p(1)+p(0)  = 2    as P(2) = set { 1+1, 2 }
p(1) = 1      as  P(1) = {1 }
p(0) = 1    by definition
p(i) = 0  for all i negative.

==================
p(12) = p(11) + p(10) - p(7) - p(5) + p(0)
= 56 + 42 - 15 - 7 + 1
= 77

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2015-05-20T20:41:41+05:30
The answer seems to be the sum of   1, 1, 2, 3, 5, 7, 11, 11, 10,  5, 1  =  57    and in standard websites, the p(11) is given as 56.

11
10 + 1
9 + 1 + 1      or  9 + 2
8 + 1 + 1 + 1    or  8 + 1 + 2   or 8 + 3
7 + 1 + 1 + 1 + 1  or  7 + 2 + 1 + 1  or  7 + 2 + 2  or  7 + 3 + 1  or  7 + 4

6+1+1+1+1+1, 6+2+1+1+1, 6+2+2+1, 6+3+2, 6+3+1+1,6+4+1, 6+5

5+1+1+1+1+1+1,  5+2+1+1+1+1, 5+2+2+1+1, 5+2+2+2, 5+3+1+1+1,
5+3+2+1, 5 +3+3, 5+4+1+1, 5+4+2, 5+5+1, 5+6

4+1+1+1+1+1+1+1, 4+2+1+1+1+1+1, 4+2+2+1+1+1, 4+2+2+2+1,
4+3+1+1+1+1, 4+3+2+1+1, 4+3+2+2, 4+3+3+1, 4+4+1+1+1, 4+4+2+1,
4+4+3,           we haven’t  4+5+2, 4+5+1+1, 4+6+1, 4+7

3+1+1+1+1+1+1+1, 3+2+1+1+1+1+1+1, 3+2+2+1+1+1+1, 3+2+2+2+1+1,
3+2+2+2+2,  3+3+1+1+1+1+1, 3+3+2+1+1+1, 3+3+2+2+1, 3+3+3+2, 3+3+3+1+1,
we haven’t 4+4+3, 4+3+1+1+1+1, 4+3+2+1+1, 4+3+2+2, 4+3+3+1, 5+3+1+2,
5 +3+3, 5+3+1+1+1, , 6+3+2, 6+3+1+1, 7+3+1, 8+3

2+2+2+2+2+1,2+2+2+2+1+1+1, 2+2+2+1+1+1+1+1, 2+2+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1,

1+1+1+1+1+1+1+1+1+1+1

p(11) = 56 or 57   by enumerating all possibilities.
==========================

By partition theorem:
g(i) = (3 i² - i) / 2   = Euler's pentagonal number

for all i  positive and negative except 0.

g(-4) = 25,   g(-3) = 15,  g(-2) = 7,  g(-1) = 2 ,  g(1) = 1,  g(2) = 5,  g(3) = 12,  g(4) = 22

Partition(11) =  ...- Part(11-25) + Part(11-15) - Part(11-7) + Part(11 -2) + part(11-1) - Part(11-5) + part(11-12) - Part(11-22) + ....
Partition (11)  =  0+...+0 + 0 - Part(4) + Part(9) +Part(10) - Part(6) + 0 - 0 +...

p(11) = p(9) + p(10) - p(4) - p(6) = 56

p(10) = p(9) + p(8) - p(3) - p(5) = 42
p(9) = p(8) + p(7) - p(4) - p(2) = 30
p(8) = p(7)+p(6)- p(3)- p(1) = 22
p(7) = p(6)+p(5) - p(2) - p(0) =  15
p(6) = p(5)+p(4) - p(1) = 11
p(5) = p(4+p(3) - p(0) = 7
p(4) = p(3)+p(2) = 5      as P(4) = {1+1+1+1, 1+2+1, 2+1, 3+1 ,4}
p(3) = p(2)+p(1) = 3      as P(3) = { 1+1+1 , 1+2 , 3}
p(2) = p(1)+p(0)  = 2    as P(2) = set { 1+1, 2 }
p(1) = 1      as  P(1) = {1 }
p(0) = 1    by definition
p(i) = 0  for all i negative.

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p(12) = p(11) + p(10) - p(7) - p(5) + p(0)
= 56 + 42 - 15 - 7 + 1
= 77