Monomial symmetric functions

class sage.combinat.sf.monomial.SymmetricFunctionAlgebraElement_monomial(A, x)
expand(n, alphabet='x')

Expands the symmetric function as a symmetric polynomial in n variables.

EXAMPLES:

sage: m = SFAMonomial(QQ)
sage: m([2,1]).expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2
sage: m([1,1,1]).expand(2)
0
sage: m([2,1]).expand(3,alphabet='z')
z0^2*z1 + z0*z1^2 + z0^2*z2 + z1^2*z2 + z0*z2^2 + z1*z2^2
sage: m([2,1]).expand(3,alphabet='x,y,z')
x^2*y + x*y^2 + x^2*z + y^2*z + x*z^2 + y*z^2
class sage.combinat.sf.monomial.SymmetricFunctionAlgebra_monomial(R)
__init__(R)

TESTS:

sage: m = SFAMonomial(QQ)
sage: m == loads(dumps(m))
True
_multiply(left, right)

EXAMPLES:

sage: m = SFAMonomial(QQ)
sage: a = m([2,1])
sage: a^2
4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
sage: QQx.<x> = QQ['x']
sage: m = SFAMonomial(QQx)
sage: a = m([2,1])+x
sage: 2*a # indirect doctest
2*x*m[] + 2*m[2, 1]
sage: a^2
x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
dual_basis(scalar=None, scalar_name='', prefix=None)

The dual basis of the monomial basis with respect to the standard scalar product is the homogeneous basis.

EXAMPLES:

sage: m = SFAMonomial(QQ)
sage: h = SFAHomogeneous(QQ)
sage: m.dual_basis() == h
True

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