Help:=proc(): print(`BonnSieve(n,k), `):
print(` ApplySieve(Se,P,Si)`): end:
with(combinat):

#{{set}}: A collection of subsests of {1,2, ....n}
#such that Sum((-1)^s*ElementsWithProperties[s], s in the 
#{set} gives a lower or upper bounds (if k is even or odd)
#(either resp. or not)
BonnSieve:=proc(n,k) local i,N:
N:={seq(i,i=1..n)}:
{seq(op(choose(N,i)),i=0..k)}:

end:


#ApplySieve(Se,P,Si): Inputs a universal se Se
#and a list of "properties" described explicitly
#by a list of subsets of Se P=[P1,P2, ...]
#(meaning Pi is the set of objectts having property i
#(and possiblly other properties), and it also inputs
# a sieve Si. Outputs the "estimate" implied by Si
#and the true value of the set of objects that has
#none of the properties
#For example
#ApplySieve({Emilie,Aisha,Daniela},
#[{Emilie,Aisha},{Emilie,Daniela}], {{},{1},{2}})
#should return
#TrueValue=0
#Estimate=3-2-2=-1
ApplySieve:=proc(Se,P,Si)  local Es, TV,josh,kellen,Si1:

if not {} in Si then
 ERROR(`Bad input`):
fi:

TV:=nops(Se minus {seq(op(P[i]),i=1..nops(P))}):

Si1:=Si minus {{}}:


Es:=add((-1)^nops(josh)*nops(`intersect`(seq(P[kellen],
kellen in josh))),josh in Si1)  +  nops(Se):
Es,TV:

end:


#k=Number of ordered properties
#BrunSieve(k,N) inputs the number of properties k
#and a list N of weakly decrreasing pos. integers
#N=[N1,N2,N3, Nr] with r<=k
#outputs all subsets of {1, ..., k} written
#in dec. order i1>i2>i3>.. such that
#i1<=N1 and i2<=N2, ...ir<=Nr  
BrunSieve:=proc(k,L)
end: