def METHOD_NAME(self, r):
    r"""
    Give the `r^{th}` homogeneous function inside the Nil-Coxeter algebra.
    In finite type `A` this is the sum of all decreasing elements of length `r`.
    In affine type `A` this is the sum of all cyclically decreasing elements of length `r`.
    This is only defined in finite type `A`, `B` and affine types `A^{(1)}`, `B^{(1)}`, `C^{(1)}`, `D^{(1)}`.
    INPUT:
    - ``r`` -- a positive integer at most the rank of the Weyl group
    EXAMPLES::
        sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
        sage: U.homogeneous_generator_noncommutative_variables(2)
        u[1,0] + u[2,0] + u[0,3] + u[3,2] + u[3,1] + u[2,1]
        sage: U = NilCoxeterAlgebra(WeylGroup(['B',4]))
        sage: U.homogeneous_generator_noncommutative_variables(2)
        u[1,2] + u[2,1] + u[3,1] + u[4,1] + u[2,3] + u[3,2] + u[4,2] + u[3,4] + u[4,3]
        sage: U = NilCoxeterAlgebra(WeylGroup(['C',3]))
        sage: U.homogeneous_generator_noncommutative_variables(2)
        Traceback (most recent call last):
        ...
        AssertionError: Analogue of symmetric functions in noncommutative variables is not defined in type ['C', 3]
    TESTS::
        sage: U = NilCoxeterAlgebra(WeylGroup(['B',3,1]))
        sage: U.homogeneous_generator_noncommutative_variables(-1)
        0
        sage: U.homogeneous_generator_noncommutative_variables(0)
        1
    """
    ct = self._cartan_type
    msg = f"Analogue of symmetric functions in noncommutative variables is not defined in type {ct}"
    assert (len(ct) == 2 and ct[0] in ['A', 'B']) or (len(ct) == 3 and ct[2] == 1), msg
    if r >= self._n:
        return self.zero()
    return self.sum_of_monomials(w for w in self._W.pieri_factors() if w.length() == r)