def _from_Monomial_on_basis(self, J):
r"""
Given a quasi-symmetric monomial function, this method returns the expansion into the dual immaculate basis.
INPUT:
- ``J`` -- a composition
OUTPUT:
- A quasi-symmetric function in the dual immaculate basis.
EXAMPLES:
sage: dI = QuasiSymmetricFunctions(QQ).dI()
sage: dI._from_Monomial_on_basis(Composition([]))
dI[]
sage: dI._from_Monomial_on_basis(Composition([2,1]))
-dI[1, 1, 1] - dI[1, 2] + dI[2, 1]
sage: dI._from_Monomial_on_basis(Composition([3,1,2]))
-dI[1, 1, 1, 1, 1, 1] + dI[1, 1, 1, 1, 2] + dI[1, 1, 1, 3] - dI[1, 1, 4] - dI[1, 2, 1, 1, 1] + dI[1, 2, 3] + dI[2, 1, 1, 1, 1] - dI[2, 1, 1, 2] + dI[2, 2, 1, 1] - dI[2, 2, 2] - dI[3, 1, 1, 1] + dI[3, 1, 2]
"""
n = J.size()
C = Compositions()
C_n = Compositions(n)
mat = self._matrix_monomial_to_dual_immaculate(n)
column = C_n.list().index(J)
return self.sum_of_terms( (C(I), mat[C_n.list().index(I)][column])
for I in C_n)
