#OK to post homework
#Michael Yen, 10/18/20, Assignment 12
#1) By iterating procedure CP(L1,L2), in
#M12.txt

#write a procedure

#CPg(L)

#that inputs a list of lists L=[L1,L2, ...,Lk] of "databases" (represented as lists L1, L2, ..., Lk) and outputs

#CP(L1,L2, ..., Lk)

#In particular, CPg([L1,L2]) should output the same as CP(L1,L2) for any two lists of integers L1 and L2

CPg:=proc(L) local i,j,n,sum,L2: option remember:
L2:=[]:
n:=nops(L):
for i from 1 to n do:
 for j from i+1 to n do:
  L2:=[op(L2),op(CP(L[i],L[j]))]:
 od:
od:

L2:
end:

#2) Procedures AveClever(L) and kthMomentClever(L,k) inputs a database L, and first finds its weight-enumerator. #Suppose that you already know the weight-enumerator, call it f, in the variable x. Modify these procedures to 
#write procedures
#AveGF(f,x) and kthMomentGF(f,x,k)

#that do the same things if you already have the weight-enumerator f (in terms of x).

#In particular, for any "data-base" L, check that

#AveGF(WtEn(L,x),x)= AveClever(L)

#and

#kthMomentGF(WtEn(L,x),x,k)= kthMomentClever(L,k)

AveGF:=proc(f,x) option remember:
subs(x=1,diff(f,x))/subs(x=1,f):
end:

kthMomentGF:=proc(f,x,k) local mu,f1,i: option remember:
mu:=AveGF(f,x):
f1:=f/x^mu/subs(x=1,f):
for i from 1 to k do
f1:=expand(x*diff(f1,x)):
od:
subs(x=1,f1):
end:

#3) The scaled moment about the mean of a "random variable" X (in other words some numerical attribute on a #combinatorial) set, defined on a "sample spaces" is defined by
#m_k(X)/m_2(X)^(k/2)

#, where m_k(X) is the k-th moment about the mean.

#Write a procedure

#ScaledMomentGF(f,x,k)

#that compute it, given the weight-enumerator (alias generating-function)

ScaledMomentGF:=proc(f,x,k) local m_k,m_2: option remember:
m_k:=kthMomentGF(f,x,k):
m_2:=kthMomentGF(f,x,2):
m_k/m_2^(k/2):
end:

#4) We proved in class that the weight-enumerator of the set of words in the alphabet {0,1} of length n, according 
#to their sum (equivalently, the number of 1-s) is (1+x)n
#Leaving n as a symbol, use Maple to find explicit expressions for

#ScaledMomentGF(((1+x)/2)^n,x,k), k=2,3,4,5,6,7,8,9,10

#Then take limits as n goes to infinity (using the Maple command lim( ..., n=infinity)) Get a sequence of numbers. #What is it?

#Is it in the OEIS?

#[seq(limit(ScaledMomentGF(((1 + x)/2)^n, x, k), n = infinity), k = 2 .. 10)];
#[1, 0, 3, 0, 15, 0, 105, 0, 945]

#Yes, this is in the OEIS with A number A123023.