#OK to post homework
#William Wang, 12/10/2020, Assignment 25

#1.
#Draw the Ferrers diagram of the partition [9,6,5,3,1,1,1]. What is it's conjugate partition?
Ferrers diagram of the partition [9,6,5,3,1,1,1]

* * * * * * * * *
* * * * * *
* * * * *
* * *
*
*
*

Its conjugate partition (shown below): [7,4,4,3,3,2,1,1,1]

* * * * * * *
* * * *
* * * *
* * *
* * *
* *
*
*
*



#2.
#How many partitions of 151 are there with exactly 10 parts?
#Theorem: The number of partitions of n whose largest part is k is equal to the number of partitions of n into k parts.
#So, the number of partitions of 151 into exactly 10 parts is equal to the number of partitions of n whose largest part is k.
pnk(151, 10);
  79811865

#There are 79811865 partitions of 151 with exactly 10 parts.



#3.
By hand: [[6,4,2,2,2,2,2],0]

BZ([[6, 5, 3, 1, 1, 1, 1, 1], -1]);
  [[6, 4, 2, 2, 2, 2, 2], 0]

#They agree



#4.
#If you want to find the number of partitions of 10000 using procedure pnFast and type pnFast(10000); you would get an error message. Why?
I did not get an error message when doing pnFast(10000). 

#Use this way to find the number of partitions of 20000.
[seq(pnFast(i), i = 1 .. 20000)][20000];
  252114813812529697916619533230470452281328949601811593436850314108034284423801564956623970731689824369192324789351994903016411826230578166735959242113097

#How far can you get?
#Was able to get pnFast(100000) in a reasonable amount of time:
[seq(pnFast(i),i=1..100000)][100000]
  27493510569775696512677516320986352688173429315980054758203125984302147328114964173055050741660736621590157844774296248940493063070200461792764493033510116079342457190155718943509725312466108452006369558934464248716828789832182345009262853831404597021307130674510624419227311238999702284408609370935531629697851569569892196108480158600569421098519



#5.
pnFastMod := proc(n, m):
pnFast(n) mod m:
end:

pnFastMod(10^5, 101);
  89