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

1
by sriramramu
is this itself is the question

2015-04-21T18:06:52+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
The partitions of 11 are:
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+1+2, 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,
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,
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