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simplifying complex polynomials

A friend of mine is trying to simplify polynomials which are sums of
monomials each of which is the product of terms which are y_[] and z_[]
where the [] is a vector like [1,1,2,1,5].

Her simplification rules are fairly straightforward -- but how do you
get Mathematica to do it.

Rule 1: Each monomial must be quadratic in z (i.e. if it's not, replace
it by 0)

Rule 2: Each monomial's vector subscripts must sum (over all z's and
y's) to the vector [1,2,2,2,1]  (for example). If it doesn't the term
can be eliminated.

Any thoughts? She's trying to process it as a list but this seems to be
further complicated by multiplicities.

Thanks, Daniel

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