Rigid Hilbert Polynomials for m-Primary Ideals

Which Hilbert polynomials for an m-primary ideal I in a d-dimensional, (d > 0), local Cohen-Macaulay ring (R, m) determine Heilbert function of I? For example, if we denote the Hilbert function giving the length of R/Inby H _I (n) and the corresponding polynomial by p _I (X), then any m-primary ideal I having Hilbert polynomial $$p_I(X) = \lambda \left( {X + \mathop d\limits_d - 1} \right)$$, has Hilbert function $$H_I = \lambda \left( {n + \mathop d\limits_d - 1} \right)$$ for all n > 0 and, in addition, I must be generated by d elements.