#OK to post homework
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Q1. Garvan Maple booklet (first ~30 pages): small transcript sample
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# Basic exact arithmetic vs floating point
> 105/25;
                             21/5
> evalf(105/25);
                          4.200000000
> evalf(Pi,30);
                   3.14159265358979323846264338328

# Assignment and case-sensitivity
> f := 21/5:
> G := -1/5:
> f+g;
                           21/5 + g
> f+G;
                               4

# Algebra commands: expand / factor / simplify / collect
> expand((x+2)^5);
      x^5 + 10*x^4 + 40*x^3 + 80*x^2 + 80*x + 32
> factor(x^4-1);
                     (x - 1) (x + 1) (x^2 + 1)
> simplify((x^2-1)/(x-1));
                             x + 1

# Radicals
> simplify(sqrt(72));
                           6*sqrt(2)
> radsimp(sqrt(50)+sqrt(8));
                           7*sqrt(2)

# Coefficients + substitution
> coeff(expand((x+1)^7), x, 3);
                              35
> subs(x=3, (x^2+1)/(x-1));
                               5

# Solving equations
> solve(x^2-5*x+6=0, x);
                             2, 3
> solve({x+y=1, x-y=3}, {x,y});
                       {x = 2, y = -1}

# A small loop (od) + if (fi) example
> S := 0:
> for i from 1 to 5 do
>     S := S + i^2;
> od:
> S;
                              55

> absdiff := proc(a,b)
>     if a>=b then
>         a-b;
>     else
>         b-a;
>     fi;
> end:
> absdiff(3,10);
                               7


Q2. Go over today's Maple code and try to understand it. - Done


Q3. Do a preliminary reading of Dr. Z.'s summary of Enumeartive and Algebraic combinatorics - Done



Q4. Procedure NuFP(pi)
----------------------
# NuFP(pi) = number of fixed points of permutation pi, i.e. # {i : pi[i]=i}.

NuFP := proc(pi)
local n, i, c;

n := nops(pi);
c := 0;

for i from 1 to n do
    if pi[i] = i then
        c := c + 1;
    fi;
od;

return c;
end:




Q5. Procedure WtE(n,x) and OEIS identification
----------------------------------------------
# WtE(n,x) = sum_{pi in S_n} x^NuFP(pi).

with(combinat):

WtE := proc(n,x)
local S, pi;

if n=0 then
    return 1;   # empty permutation, NuFP=0, so weight = x^0 = 1
fi;

S := 0;
for pi in permute(n) do
    S := S + x^NuFP(pi);
od;

return expand(S);
end:

# Example:
> WtE(4,x);
                      x^4 + 6*x^2 + 8*x + 9


# For k = 0..5, compute sequences:
# [seq(coeff(WtE(i,x), x, k), i=0..8)];
#
# Results (n=0..8):
#
# k=0: [1, 0, 1, 2, 9, 44, 265, 1854, 14833]
# k=1: [0, 1, 0, 3, 8, 45, 264, 1855, 14832]
# k=2: [0, 0, 1, 0, 6, 20, 135, 924, 7420]
# k=3: [0, 0, 0, 1, 0, 10, 40, 315, 2464]
# k=4: [0, 0, 0, 0, 1, 0, 15, 70, 630]
# k=5: [0, 0, 0, 0, 0, 1, 0, 21, 112]
#
# OEIS A-numbers (rencontres numbers, fixed points of permutations):
# k=0 -> A000166  
# k=1 -> A182390  
# k=2 -> A000387 
# k=3 -> Not Exist (have significant overlap with A000449)
# k=4 -> Not Exist (overlap with A000475)
# k=5 -> Not Exist

Q6. 
(end)