On polynomial-like functions

A polynomial-like function (PLF) of degree n is a smooth function F whose nth derivative never vanishes. A PLF has ⩽n real zeros; in case of equality it is called hyperbolic; F(i) has ⩽n−i real zeros. We consider the arrangements of the n(n+1)/2 distinct real numbers xk(i), i=0,…,n−1, k=1,…,n−i, which satisfy the conditions xk(i)