linear algebra – second derivative nilpotent index

I have a linear operator on the complex field that is defined as the second derivative of x and y $$ frac { partial} { partial x ^ 2} + frac { partial} { partial y ^ 2} $$ V is the vector space of all the polynomials of the variables x and y with maximum degree n with complex coefficients. I want to find the index of this operator. I know that the degree will be less than the dimension of the operator, but it seems that it should be a lower upper limit that is somehow related to the "middle" exponents of the binomial expansion of (x + y) ^ n (eg. the operator is nilpotent with an index of max 2 for polynomials of degree 4). But I don't know how to describe this in a better way. Any help would be appreciated.