set up permutahedron with precomputed data

src/sage/geometry/polyhedron/library.py

@@ -2212,11 +2212,35 @@ def permutahedron(self, n, project=False, backend=None):
 
             sage: p4 = polytopes.permutahedron(4,backend='normaliz')   # optional - pynormaliz
             sage: TestSuite(p4).run()                                  # optional - pynormaliz
+
+        Check that precomputed data is correct::
+
+            sage: P = polytopes.permutahedron(5, backend='field')
+            sage: assert P == Polyhedron(P.vertices())
+            sage: assert P == Polyhedron(ieqs=P.inequalities(), eqns=P.equations())
         """
-        verts = list(itertools.permutations(range(1, n + 1)))
+        verts = tuple(itertools.permutations(range(1, n + 1)))
         if project:
             verts = project_points(*verts)
-        return Polyhedron(vertices=verts, backend=backend)
+            return Polyhedron(vertices=verts, backend=backend)
+        else:
+            parent = Polyhedra(ZZ, n, backend=backend)
+            def tri(m):
+                return (m*(m+1))//2
+
+            # Each proper `S \subset [n]` corresponds exactly to
+            # a facet that minimizes the coordinates in `S`.
+            # The minimal sum for `m` coordinates is `(m*(m+1))/2`.
+            ieqs = tuple((-tri(sum(x)),) + x
+                         for x in itertools.product([0,1], repeat=n)
+                         if 0 < sum(x) < n)
+
+            # Adding the defining equality.
+            eqns = ((-tri(n),) + tuple(1 for _ in range(n)),)
+
+            return parent([verts, [], []], [ieqs, eqns],
+                          Vrep_minimal=True, Hrep_minimal=True)
+
 
     def generalized_permutahedron(self, coxeter_type, point=None, exact=True, regular=False, backend=None):
         r"""