> #Attendence Q1.
> #Use the Lagrange Inversion Formula to find an explicit expression for the
> coeff. of x^n in the power series of u(x) that satisfies the functional
> equations
> 
> #i. u(x)=x*(1+u(x))^2=x*(1+2*u(x)+u(x)^2) -> PHI(z) = 1+2*z+z^2
> #[seq(coeff(taylor(PHI^n,z=0,n),z,n-1)/n,n=1..N)]:
> #ii. u(x)=x*(1+u(x))^3
> #[seq(coeff(taylor(((1+z)^3)^n,z=0,n),z,n-1)/n,n=1..N)]:
> #iii. u(x)=x*(1+u(x))^k
> #[seq(coeff(taylor(((1+z)^k)^n,z=0,n),z,n-1)/n,n=1..N)]:
> 
> #Attendence Q2.
> read `M23.txt`;
> RandF(16);
[13, 7, 15, 10, 13, 16, 6, 2, 11, 12, 4, 6, 5, 11, 16, 1]
> #Consider the function given above,
> #f(1)=13, f(2)=7, ... , f(16)=1
> #i. draw this directed graph
> #only one cycle found
> # 14
> # 11<-9
> # 4
> # 10
> # 12
> # |
> #8->2->7->6->16->1->13<->5
> # |
> # 15
> # 3
> 
> #ii. Find the doubly rooted tree that is outputted by the joyal bijection.
> check that it as below.
> Joyal([13, 7, 15, 10, 13, 16, 6, 2, 11, 12, 4, 6, 5, 11, 16, 1]);
{{1, 13}, {1, 16}, {2, 7}, {2, 8}, {3, 15}, {4, 10}, {4, 11}, {5, 13}, {6, 7},
{6, 12}, {6, 16}, {9, 11}, {10, 12}, {11, 14}, {15, 16}}, [13, 5]
> # 14
> # 11<-9
> # 4
> # 10
> # 12
> # |
> #8->2->7->6->16->1->13*<->5**
> # |
> # 15
> # 3
> 
> #Attendence Q3.
> #Find the purffer code of length 11 of the original tree.
> #[2,4,11,8,7,6,3,6,3,2]
> 
> 
> 
> g:={{1,2},{2,3},{2,13},{3,6},{3,10},{4,5},{4,11},{6,7},{6,12},{7,8},{8,9},{10,
> 11}};
g := {{1, 2}, {2, 3}, {2, 13}, {3, 6}, {3, 10}, {4, 5}, {4, 11}, {6, 7}, {6,
12}, {7, 8}, {8, 9}, {10, 11}}
> Pruffer(g);
[2, 4, 11, 8, 7, 6, 10, 3, 6, 3, 2]
> 
> 
> 
> 
> 
>