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# Name:Treasa Bency Biju Jose
# Date: 09-24-2020
# Assignment #6
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I. 
PIE is used as an evaluation of two ways to find the cardinality of the intersection. 
This method best helps in proof and not in calculations.

evalb([seq(Pnx(n, x), n = 1 .. 7)] = [seq(PnxF(n, x), n = 1 .. 7)]);
                              true

evalb(time([seq(PnxF(n, x), n = 1 .. 8)]) < time([seq(Pnx(n, x), n = 1 .. 8)]));
                              true


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II.
PnxF(n,x) is used to find the polynomial in x of degree n whose coefficients of 
x^i is the number of permutations with exactly i fixed points. 
D(n) helps finding the number of derrangements of n objects. 
By excluding D(n) we can find the coeeficients for PnxF(n,x)
Hence by the inclusion-exclusion principle, D(n) is used as a constant term in PnxF(n,x)

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III. 
when i=		  0	1     2     3  4  5  6  7  8  9  10   11       12
OEIS A number: A000166, 0, A000387, 0, 0, 0, 0, 0, 0, 0, 0, A010054, A010815
The largest value of i for which there is an OEIS sequence is i= 12
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IV. 
d(n) = n * d(n-1) + (-1)^(n-1) for n=1,2,..
d(n-1) = (n-1) * d(n-2) + (-1)^(n-2) for n=2,3,..
d(n-2) = (n-2) * d(n-3) + (-1)^(n-3) for n=3,4,..

These recurrence functions are interlinked because, in d(n) it recurrsively calls d(n-1)
and similarly d(n-1) recurrsively calls d(n-2). 
d1 := proc(n) option remember; n*d(n - 1) + (-1)^(n - 1); end proc;
d2 := proc(n) option remember; (n - 1)*d(n - 2) + (-1)^(n - 2); end proc;
d3 := proc(n) option remember; (n - 2)*d(n - 3) + (-1)^(n - 3); end proc;

seq(d1(i), i = 1 .. 10);
       2, -1, 4, 7, 46, 263, 1856, 14831, 133498, 1334959

seq(d2(i), i = 2 .. 10);
           2, -1, 4, 7, 46, 263, 1856, 14831, 133498

seq(d3(i), i = 3 .. 10);
               2, -1, 4, 7, 46, 263, 1856, 14831


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