#Please do not post homework
#Guy Adami, 2026-02-22, Assignment 9


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with(combinat)

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#RandL(n,k): A random list of subsets of {1, ...,n} of length k

RandL:=proc(n,k) local i:
	[seq( randcomb(n, rand(2..n-2)()), i=1..k)]:
end:

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#PIEd(n,L): What the PIE computes, done directly
PIEd:=proc(n,L) local i:
	nops({seq(i,i=1..n)} minus {seq(op(L[i]),i=1..nops(L))}):
end:

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IntL:=proc(L) local i:
	`intersect`(seq(L[i],i=1..nops(L))):
end:

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#PIE(n,L):  
PIE:=proc(n,L) local k,cu,S,s:
	cu:=n:
	for k from 1 to nops(L) do
 		S:=choose(L,k):
 		cu:=cu+(-1)^k*add(nops(IntL(s)), s in S):
	od:
	cu: 
end:

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#PIEg(n,L,t):  The weight-enumerator of the members of the universal set
#according to t^#NumberOfProperties
PIEg:=proc(n,L,t) local k,cu,S,s:
	cu:=n:
	for k from 1 to nops(L) do
 		S:=choose(L,k):
		cu:=cu+(t-1)^k*add(nops(IntL(s)), s in S):
	od:
	expand(cu): 
end:

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#Question 1

NumberOfProperties:=proc(n,L,i) local counter, j, k:
	counter:= 0:
	for j from 1 to nops(L) do
		for k from 1 to nops(L[j]) do
			if i = L[j][k] then
				counter:= counter + 1:
			fi:
		od:
	od:
	counter;
end:

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#Question 2


PIEgD := proc(n, L, t) local T, i, j, NumProps; 
	NumProps := []:
	for i to n do 
		NumProps := [op(NumProps), NumberOfProperties(n, L, i)]; 
	od: 
	T := sum(t^NumProps[j], j = 1 .. n); 
	T; 
end:

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#Question 3

#L:=RandL(1000,4):
#evalb(PIEg(1000,L,t)=PIEgD(1000,L,t));

#This evaluation held true all 5 times