Polynomials

Consider a polynomial with $k$ variables, represented as a sum of monomials:

,

Here, p(i, j) denotes the degree of the $j$-th variable in the $i$-th monomial (p(i, j) \geq 0), and a_i is a non-zero integer constant. The degree of a monomial is the sum of the degrees of all its variables. The degree of a polynomial in several variables is the maximum degree among all its monomials, expressed as an integer. By minimizing $m$ (after combining like terms) and ordering the monomials by some criterion, for example, first by degree and then lexicographically, we obtain the canonical representation of the polynomial. This means that any polynomial will have a unique expression in this representation.

A polynomial is called complete if its canonical representation contains all possible monomials. For example, the complete polynomial of degree $3$ in $2$ variables is:

P(x, y) = a_{10}x^3 + a_9x^2y + a_8xy^2 + a_7y^3 + a_6x^2 + a_5xy + a_4y^2 + a_3x + a_2y + a_1

Once, little Dima started studying the very complex topological properties of algebraic structures, and faced the following problem.

Dima has a complete polynomial of degree $n$ in $k$ variables and now he is interested in the following:

(a) If the degree of the polynomial is even, how many monomials of even degree are present in its canonical representation?

(b) If the degree of the polynomial is odd, how many monomials of odd degree are present in its canonical representation?