# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
# have to explicitly mangle our sitedir here so that "mjo.cone"
# resolves.
from os.path import abspath
from site import addsitedir
addsitedir(abspath('../../'))

from sage.all import *

def rearrangement_cone(p,n):
    r"""
    Return the rearrangement cone of order ``p`` in ``n`` dimensions.

    The rearrangement cone in ``n`` dimensions has as its elements
    vectors of length ``n``. For inclusion in the cone, the smallest
    ``p`` components of a vector must sum to a nonnegative number.

    For example, the rearrangement cone of order ``p == 1`` has its
    single smallest component nonnegative. This implies that all
    components are nonnegative, and that therefore the rearrangement
    cone of order one is the nonnegative orthant.

    When ``p == n``, the sum of all components of a vector must be
    nonnegative for inclusion in the cone. That is, the cone is a
    half-space in ``n`` dimensions.

    INPUT:

    - ``p`` -- The number of components to "rearrange."

    - ``n`` -- The dimension of the ambient space for the resulting cone.

    OUTPUT:

    A polyhedral closed convex cone object representing a rearrangement
    cone of order ``p`` in ``n`` dimensions.

    EXAMPLES:

    The rearrangement cones of order one are nonnegative orthants::

        sage: rearrangement_cone(1,1) == Cone([(1,)])
        True
        sage: rearrangement_cone(1,2) == Cone([(0,1),(1,0)])
        True
        sage: rearrangement_cone(1,3) == Cone([(0,0,1),(0,1,0),(1,0,0)])
        True

    When ``p == n``, the resulting cone will be a half-space, so we
    expect its lineality to be one less than ``n`` because it will
    contain a hyperplane but is not the entire space::

        sage: rearrangement_cone(5,5).lineality()
        4

    TESTS:

    todo.
    should be permutation invariant.
    should have the expected lyapunov rank.
    just loop through them all for n <= 10 and p < n?

    """

    def d(j):
        v = [1]*n    # Create the list of all ones...
        v[j] = 1 - p # Now "fix" the ``j``th entry.
        return v

    V = VectorSpace(QQ, n)
    G = V.basis() + [ d(j) for j in range(n) ]
    return Cone(G)