I am working on a dissertation in math at Rutgers University. My goal is to find as simple a formulae as possible for the linear, differential equation of order N with x as dependent variable, y as independent variable, and y=P(x) is a polynomial in x of degree N, P(0)=0. This linear ODE is called the "differential resolvent" (DR) of the polynomial, P. The coefficient functions making up DR are polynomials in y and the coefficients of P. The gcd of these polynomials is 1.
In the course of this work, I need many steps. Here are two of them: I hope these are not elementary combinatorical problems.
1) Given positive integers n and k, how many partitions of n with no part bigger than k exist?
2) Given partitions, P and Q, how many domino (brick) tabloids are there of shape P and type Q, as a function of the parts of P and Q?