#OK to post homework

#Zidong Zhang, 28/2/2021, Assignment 11



#RandPnk(n,k):that genearte unoformly at random, a partion of n with largest part k,


RandPnk := proc(n,k) local k1,pn1:

if k>n then
	RETURN([]):
fi:

if n = k then
	RETURN([n]):
fi:

k1:= rand(1..(n-k))():
pn1:=RandPnk(n-k, k1):
RETURN([op(pn1),k]):
end:



#RandPn(n):that genearte, unoformly at random,  a partion of n



RandPn := proc(n) local die,k:

die:= [seq(pnk(n,i), i = 1 ..n )]:
k := LD(die):
RandPnk(n,k):

end:


#EstStatAnalOfLargestPart(n,K,N): that inputs a positive integer n and a fairly large K, and (using previous functions), estimaes the expectation, variance, and the 3rd- through K-th scaled moments, of the r.v. "number of parts" when running over the sample space of (integer) partitions of n.



########I'm guessing that you mean we repeat it K times and from 3 to N-th scaled moments. But I'm not sure, but I shall try this direction.

EstStatAnalOfLargestPart := proc(n ,K ,N)  local mu , var, L, X:

X := [seq(nops(RandPn(n)), i = 1 .. K)]:

mu := Ave(X):

var:=Mom(X,2):

L := [mu,var]:

if var = 0 then 
	RETURN(L):
fi:

[op(L) , seq(evalf(Mom(X,i)/var^(i/2)) , i =3 .. N) ]
	
end:

## I run the program several times and find that the mean is about 5.93 
# and the variance is about 3.27 they are very close to each other.

####when I run EstStatAnalOfLargestPart(200,10000,6), the 3rd moment and the 4th moment is has more fluctuation than mean and variance, especially 4th moment.


#pnSeqGFt(N,t) : that inputs N, and a variable t, and outputs the first N+1 (starting at n=0) of the sequence {Pn(t)} mentioned in the previous problem.


pnSeqGFt := proc(N,t) local L , i , p ,Pn1 :

L:=[1]:

for i from 1 to N do
	Pn1:= Pn(i):
	p := add(t^nops(Pn1[j]), j =1 .. nops(Pn1)):
	L:=[op(L) , p]:
od:
L:
end:





######################from C7 and C11



LD:=proc(L) local n,i,su,r:
n:=nops(L):
r:=rand(1..convert(L,`+`))():
su:=0:

for i from 1 to n do
 su:=su+L[i]:
  if r<=su then
    RETURN(i):
  fi:
od:
end:


pnk:=proc(n,k) local k1:
option remember:
if not (type(n,integer) and type(k,integer) and n>=1 and k>=1 )then
 RETURN(0):
fi:
if n=k then
 RETURN(1):
else
 RETURN(add(pnk(n-k,k1),k1=1..min(k,n-k))):
fi:
end:


Ave:=proc(X) local i:add(X[i],i=1..nops(X))/nops(X):end:

Mom:=proc(X,k) local i,mu:
mu:=Ave(X):add((X[i]-mu)^k ,i=1..nops(X))/nops(X):
end: