Lecture 12: Due Oct. 18, 9:00pm. Email ShaloshBEKhad@gmail.com an attachment

hw12FirstLast.txt

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	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. 

		Answer:
			CPg:=proc(L) local list, i: option remember:
			list:=CP(L[1],L[2]):
			for i from 3 to nops(L) do:
				list:=CP(l1,i):
			od: 
			return list:
			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)

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

			kthMomentGF:=proc(f,x,k) local mu,f1,i:
			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)

		Answer:
			
			ScaledMomentGF:=proc(f,x,k) local a, b:
			a:=kthMomentGF(f,x,k):
			b:=kthMomentGF(f,x,2):
			a/(b^(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? 

		Answer:
			#[seq(limit(ScaledMomentGF(((1 + x)/2)^n, x, k), n = infinity), k = 2 .. 10)] = 1, 0, 3, 0, 15, 0, 105, 0, 945
			#Yes, OEIS is  A123023.