#ATTENDANCE QUIZ FOR LECTURE 14 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p14
#with an attachment called
#p14FirstLast.txt
#(e.g. p14DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 24, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE:6 + extra credit

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER
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1. Where and when was Herbert S. Wilf born? Look up his paper about 
   "random generation of combinatorial objects". Advance in Mathematics ca.1980 and at least 
    skim it.
A1. Herbert S. Wilf was born on June 13 1931 at Philedelphia, PA.
     Wilf's Combinatorial Algorithms, published by Academic Press in 1978, specializes in algorithms 
     for constructing basic combinatorial objects such as permutations, subsets, and partitions; both 
     randomly and sequentially.Dr. Doron Zeilberger was one of his collaboators.

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2. Let m:= age of Donald Trump
       n:= age of Joe Biden
   What is the probability that if you take a random lattice walk
   from [0,0] to [n,m] you will always stay in the region x >= y
A2. 
T := RPath(77, 74);
T := [1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 

  2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 

  1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 

  1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 

  1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 

  2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 

  1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 

  1, 1, 1, 2, 1]



DrawP(T);

(attached as file)
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3. What is a Wilf-Zeilberger pair?
A3. Wilf-Zeilberger par is instrumental in combinatorics as they certify certain combinatorial 
    identities. They help in evaluation of many sums involving binomial coefficients, factorials, 
    and any hypergeometric series. 
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4. What nationality was Catalan? What is the constant named after him?
A4. Eugene Charles Catalan was born in Belgium in 1814. Catalan constant (G) is defined by

G:= ∑k = 0 ∞ (−1)k(2k+1)2 ≈ 0.915965594177219015…

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5. Do the same for [1, 1, -1, -1, 1] and get a set of 10 such lists that is the same whole 
   of Paths(3,2);
A5. V := Paths(3, 2);
V := {[-1, -1, 1, 1, 1], [-1, 1, -1, 1, 1], [-1, 1, 1, -1, 1], 

  [-1, 1, 1, 1, -1], [1, -1, -1, 1, 1], [1, -1, 1, -1, 1], 

  [1, -1, 1, 1, -1], [1, 1, -1, -1, 1], [1, 1, -1, 1, -1], 

  [1, 1, 1, -1, -1]}


W := {[-1, -1, 1, 1, 1], [-1, 1, 1, 1, -1], [1, -1, -1, 1, 1], [1, 1, -1, -1, 1], [1, 1, 1, -1, -1]};
Y := {[-1, 1, -1, 1, 1], [-1, 1, 1, -1, 1], [1, -1, 1, -1, 1], [1, -1, 1, 1, -1], [1, 1, -1, 1, -1]};
W := {[-1, -1, 1, 1, 1], [-1, 1, 1, 1, -1], [1, -1, -1, 1, 1], 

  [1, 1, -1, -1, 1], [1, 1, 1, -1, -1]}


Y := {[-1, 1, -1, 1, 1], [-1, 1, 1, -1, 1], [1, -1, 1, -1, 1], 

  [1, -1, 1, 1, -1], [1, 1, -1, 1, -1]}


J := W union Y;
J := {[-1, -1, 1, 1, 1], [-1, 1, -1, 1, 1], [-1, 1, 1, -1, 1], 

  [-1, 1, 1, 1, -1], [1, -1, -1, 1, 1], [1, -1, 1, -1, 1], 

  [1, -1, 1, 1, -1], [1, 1, -1, -1, 1], [1, 1, -1, 1, -1], 

  [1, 1, 1, -1, -1]}



evalb(V = J);
                              true


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6. You are in a circular track in the desert. At random places there are gas containers with 
   random amounts of gas a1, a2, ..., ak such that the a1 + a2 + ... + ak gallons are exactly
   what you need to drive this track. [[0, 0.1],[0.2, 0.5], [0.7, 0.4]]
   Prove that there exists a location on the circular track such that if you start there you 
   don't run out of gas. device a method for it.
A6. Lets assume [[1,a1], [2,a2], ..., [k, ak]] where 1,2,..,k denotes the place and a1,a2,..,ak 
    denotes the gallons of gas. Lets say there are 5 stations and each station has around 2 
    gallons of gas. So, 1 -> 2 gal, 2 -> 1.8 gal, 3 -> 2.2, 4 -> 2.5, 5 -> 1.5
    If all the places are 2m apart and you need 1 gal to travel 1 m: 
    (i) Starting at place 1 will leave you stuck before you reach 2
    (ii) Starting at 2 will not reach 3 
    (iii) Starting at 3 will get you through the circular track

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