Power-sum symmetric functions

class sage.combinat.sf.powersum.SymmetricFunctionAlgebraElement_power(A, x)
_derivative(part)

Returns the ‘derivative’ of p([part]) with respect to p([1]).

EXAMPLES:

sage: p = SFAPower(QQ)
sage: a = p([2,1])
sage: a._derivative(Partition([2,1]))
p[2]
sage: a._derivative(Partition([1,1,1]))
3*p[1, 1]
_derivative_with_respect_to_p1()

EXAMPLES:

sage: p = SFAPower(QQ)
sage: a = p([1,1,1])
sage: a._derivative_with_respect_to_p1()
3*p[1, 1]
sage: a = p([3,2])
sage: a._derivative_with_respect_to_p1()
0
expand(n, alphabet='x')

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

EXAMPLES:

sage: p = SFAPower(QQ)
sage: a = p([2])
sage: a.expand(2)
x0^2 + x1^2
sage: a.expand(3, alphabet=['a','b','c'])
a^2 + b^2 + c^2
sage: p([2,1,1]).expand(2)
x0^4 + 2*x0^3*x1 + 2*x0^2*x1^2 + 2*x0*x1^3 + x1^4
sage: p([7]).expand(4)
x0^7 + x1^7 + x2^7 + x3^7
sage: p([7]).expand(4,alphabet='t')
t0^7 + t1^7 + t2^7 + t3^7
sage: p([7]).expand(4,alphabet='x,y,z,t')
x^7 + y^7 + z^7 + t^7
omega()

Returns the image of self under the Frobenius / omega automorphism.

EXAMPLES:

sage: p = SFAPower(QQ)
sage: a = p([2,1]); a
p[2, 1]
sage: a.omega()
-p[2, 1]
scalar(x)

Returns the standard scalar product of self and x.

Note that the power-sum symmetric functions are orthogonal under this scalar product. The value of \langle p_\lambda, p_\lambda \rangle is given by the size of the centralizer in S_n of a permutation of cycle type \lambda.

EXAMPLES:

sage: p = SFAPower(QQ)
sage: p4 = Partitions(4)
sage: matrix([ [p(a).scalar(p(b)) for a in p4] for b in p4])
[ 4  0  0  0  0]
[ 0  3  0  0  0]
[ 0  0  8  0  0]
[ 0  0  0  4  0]
[ 0  0  0  0 24]
class sage.combinat.sf.powersum.SymmetricFunctionAlgebra_power(R)
__init__(R)

TESTS:

sage: p = SFAPower(QQ)
sage: p == loads(dumps(p))
True

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