From edbce26685b7e62d04da8f037e51621551292225 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 11 Sep 2015 11:06:50 -0400
Subject: [PATCH] Begin work on the rearrangement cone.

---
 mjo/cone/rearrangement.py | 72 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 72 insertions(+)
 create mode 100644 mjo/cone/rearrangement.py

diff --git a/mjo/cone/rearrangement.py b/mjo/cone/rearrangement.py
new file mode 100644
index 0000000..6ea993c
--- /dev/null
+++ b/mjo/cone/rearrangement.py
@@ -0,0 +1,72 @@
+# 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)
-- 
2.44.2