> #ok to post 
> #Yifan Zhang, 11/13/2020
> 
> #Q1.
> #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!]
> #First, suppose we have A,B different combinatorial families, and there are a(n) and b(n) labeled objects of size n in A and B. Then we can have a new family C =A*B with c(n) labeled objects. And the size of a pair is the sum of the size of its total component for each pair. 
> #Then we have c(n) = sum(binomial(n,k)*a(k)*b(n-k), k=0..n)
> #Since c(n) is the set product of a(n) and b(n), then 
> #sum(c(n)*x^n/n!, n=0..infinity) = sum(sum(binomial(n,k)*a(k)*b(n-k), k=0..n)*x^n/n!, n=0..infinity) 
> #                                = sum(a(k)*x^k/k!, k=0..infinity)*sum(b(k)*x^(n-k)/(n-k!), n-k=0..infinity)
> #                                = EGF(A)* EGF(B)
> #Hence, EGF(C) = EGF(A) * EGF(B) => EGF(A*B) = EGF(A) * EGF(B)
> 
> 
> #Q2.
> #Using Maple or otherwise find the ordinary generating function and exponential generating function for
> #a(n)=n*(n-1)*(n-2) (n e 0)
> #a(0)=a(1)=a(2)=0, a(n)=n*(n-1)*(n-2) for n=3..infinity
> sum(n*(n-1)*(n-2)*x^n, n=3..infinity);
6*x^3/(x-1)^4
> #OGF(A) = 6*x^3/(x-1)^4
> 
> sum(n*(n-1)*(n-2)*x^n/n!, n=3..infinity);
x^3*exp(x)
> #EGF(A) = x^3*exp(x)
> 
> 
> 
> #Q3.
> #(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} ?
> 
> #[Reminder the egf of set-partitions is exp(exp(x)-1) ]
> #X(n)={SetPartition1, SetPartition2} 
> #X(100)
> egf1:=exp(exp(x)-1)*2:
> coeff(taylor(egf1,x=0, 101),x,100)*100!;
95170782553529667317581537682774415652727339373651222933232669275118228995784885245345448088435512613907115765121502
> 
> egf2:= exp(exp(x)-1)^2:
> coeff(taylor(egf2,x=0, 101),x,100)*100!;
86635961604145793709294621186421021577187015751602637085427283871117865337554218116225569701173834817160231371909161518471870
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