> #Weiji Zheng, Assignment 19, 11/22/2020
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> #OK TO POST HOMEWORK
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> 
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> #Q1
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> #Carefully read the section about exponential generating functions in Dr. Z.'s essay. Reproduce the proof that egf(AxB)=egf(A)*egf(B) in your own words, and understand it . [It may come up in the oral part of the Final Exam!]
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> #My Understanding:
> #For A and B, the 2 families with a(n) or b(n) labled objects, we can produce a set C by using the process A x B
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> #Denoted as C = A x B
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> #since we have a(k) * b(n-k) ways to pick typical objects, we can conclude (n)=Sum(binomial(n,k)*a(k)*b(n-k),k=0..n).
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> #By definition, egf(AxB) = egf(C) =  Sum(c(n)/n! * x^n, n=0..infinity)
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> #we put c(n) into it, get Sum(c(n)/n! * x^n, n=0..infinity) =  Sum( (Sum(binomial(n,k)*a(k)*b(n-k),k=0..n))/n! * x^n, n=0..infinity)
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> #                                                   egf(C)  =  Sum( (Sum(a(k)*b(n-k)/k!*(n-k)! * x^n, k=0..n), n=0..infinity)
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> #                                                   egf(C)  =  sum(a(k)*x^k/k!, k=0..infinity) * sum(b(k)*x^(n-k)/(n-k!), n-k=0..infinity)
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> #while [sum(a(k)*x^k/k!, k=0..infinity)] and [sum(b(k)*x^(n-k)/(n-k!), n-k=0..infinity)] are excatly egf(A) and egf(B)
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> #we get egf(C) = egf(A*B) = egf(A) * egf(B)
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> 
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> #Q2
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> #Using Maple or otherwise find the ordinary generating function and exponential generating function for a(n)=n*(n-1)*(n-2) (n ¡Ý 0)
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> #a(n)=n*(n-1)*(n-2) (n ¡Ý 0)
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> 
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> #ogf
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> #0
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> #0
> #0
> #3*2*1 = 6
> #4*3*2 = 24
> #5*4*3
> #6*5*4
> #7*6*5
> #we get a(n)=n*(n-1)*(n-2) = n!/(n-3)!
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> #fx = 
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> #sum(n*(n-1)*(n-2)*x^n, n=3..infinity); ==> 6*x^3/(x-1)^4
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> 
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> #egf
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> #sum(n*(n-1)*(n-2)*x^n/n!, n=3..infinity); ==> x^3*exp(x)
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> #Q3
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> #(i)How many ordered pairs [SP1,SP2] are there where SP1 and SP2 are set partitions of different sets whose total numbers is 100 and the labels are {1, ..., 100}
> #(ii)How many sets {SP1,SP2} where SP1 and SP2 are set-partitions of different sets whose total numbers is 100 and the labels are {1, ..., 100}?
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> #since egf of sp is exp(exp(x)-1)
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> #we can use
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> #coeff(taylor(EGF, x = 0, 101), x, 100)*100! while egf = exp(exp(x)-1) * 2 in 1 but egf = exp(exp(x)-1) * exp(exp(x)-1) in 2
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