I am very interested in the contribution of different terms in a zonal polynomial. Let's focus on the simplest case. In (20) in "Distributions of matrix variates and latent roots derived from normal samples", we have $$ Z_\kappa(I_m)=2^k(\frac{1}{2}m)_\kappa $$ Since we can write $Z_\kappa(I_m)$ into combination of monomial symmetric functions $m_\kappa(I_m)$ with positive coefficients, we may ask: how much of $Z_\kappa(I_m)$ is contributed by monomials with partition whose height is larger a given number n (that is, $\kappa_1 \geq n$)?
I think it will be impossible to get a closed-form answer. Any upper or lower bound will be extremely welcomed! If this is still difficult, we can simply focus on some special cases like m goes to infinity.
