#ATTENDANCE QUIZ FOR LECTURE 23 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p23
#with an attachment called
#p23FirstLast.txt
#(e.g. p23DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Dec. 4, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE:3

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER
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1. Use the Lagranche Inversion formula to find an explicit expression for the coefficient of x^n 
in the power series of u(x) that satisfies the functional equation 
(i) u(x) = x*(1+u(x))^2
L := [seq(x*(1 + u(x))^2, x = 1 .. 10)];
         L := [4, 162, 12, 16, 20, 24, 28, 32, 36, 40]

with(gfun);
guessgf(L, x);
              [       3        2                 ]
              [  154 x  - 308 x  + 154 x + 4     ]
              [- ---------------------------, ogf]
              [           2                      ]
              [         -x  + 2 x - 1            ]


(ii) u(x) = x*(1+u(x))^3
G := [seq(x*(1 + u(x))^3, x = 1 .. 10)];
         G := [8, 1458, 24, 32, 40, 48, 56, 64, 72, 80]

with(gfun);
guessgf(G, x);
            [        3         2                  ]
            [  1442 x  - 2884 x  + 1442 x + 8     ]
            [- ------------------------------, ogf]
            [            2                        ]
            [          -x  + 2 x - 1              ]


(iii) For any k, pos. integer u(x) = x*(1+u(x))^k 

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2. F:=RandF(16)
Consider the function given above, this means 
f(1) = 13; f(2) = 7; ...; f(16) = 1
(i) Draw this directed graph
Find the doubly-rooted tree that is outputted by the Joyal Bijection
Check that it as below 
Joyal(F)
A2. 
F := RandF(16);
    F := [2, 9, 5, 9, 9, 4, 3, 13, 15, 4, 6, 4, 4, 10, 8, 2]

L := Graph(16, directed, {{2}, {3}, {4}, {5}, {6}, {8}, {9}, {10}, {13}, {15}});
 L := Graph 6: a directed unweighted graph with 16 vertices, 0 

    arc(s), and 10 self-loop(s)




DrawGraph(L);

Joyal(F);
{{1, 2}, {2, 9}, {2, 16}, {3, 5}, {3, 7}, {4, 6}, {4, 8}, 

  {4, 10}, {4, 12}, {4, 15}, {5, 9}, {6, 11}, {9, 13}, {10, 14}, 

  {13, 15}}, [9, 8]




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3. Find the Pruffer code (of length 11) of the original tree
A3.
Pruffer(RandTree(11));
                 [9, 9, 4, 8, 11, 2, 3, 10, 7]

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