Add iterators, Whitney numbers, characteristic polynomial, etc

src/sage/matroids/basis_exchange_matroid.pxd

@@ -61,8 +61,8 @@ cdef class BasisExchangeMatroid(Matroid):
     cpdef _augment(self, X, Y) noexcept
     cpdef _is_independent(self, F) noexcept
 
-    cpdef f_vector(self) noexcept
-    cdef  _f_vector_rec(self, object f_vec, bitset_t* flats, bitset_t* todo, long elt, long rnk) noexcept
+    cpdef whitney_numbers2(self) noexcept
+    cdef  _whitney_numbers2_rec(self, object f_vec, bitset_t* flats, bitset_t* todo, long elt, long rnk) noexcept
     cpdef flats(self, R) noexcept
     cdef  _flats_rec(self, SetSystem Rflats, long R, bitset_t* flats, bitset_t* todo, long elt, long rnk) noexcept
     cpdef coflats(self, R) noexcept

src/sage/matroids/basis_exchange_matroid.pyx

@@ -1232,27 +1232,26 @@ cdef class BasisExchangeMatroid(Matroid):
 
     # enumeration
 
-    cpdef f_vector(self) noexcept:
+    cpdef whitney_numbers2(self) noexcept:
         r"""
-        Return the `f`-vector of the matroid.
+        Return the Whitney numbers of the second kind of the matroid.
 
-        The `f`-*vector* is a vector `(f_0, ..., f_r)`, where `f_i` is the
-        number of flats of rank `i`, and `r` is the rank of the matroid.
+        The Whitney numbers of the second kind are here encoded as a vector
+        `(W_0, \ldots, W_r)`, where `W_i` is the number of flats of rank `i`,
+        and `r` is the rank of the matroid.
 
-        OUTPUT:
-
-        List of integers.
+        OUTPUT: a list of integers
 
         EXAMPLES::
 
             sage: M = matroids.catalog.S8()
-            sage: M.f_vector()
+            sage: M.whitney_numbers2()
             [1, 8, 22, 14, 1]
         """
         cdef bitset_t *flats
         cdef bitset_t *todo
         if self._matroid_rank == 0:
-            return [0]
+            return [1]
         flats = <bitset_t*>sig_malloc((self.full_rank() + 1) * sizeof(bitset_t))
         todo = <bitset_t*>sig_malloc((self.full_rank() + 1) * sizeof(bitset_t))
 
@@ -1264,17 +1263,17 @@ cdef class BasisExchangeMatroid(Matroid):
         bitset_clear(todo[0])
         self.__closure(flats[0], todo[0])
         bitset_complement(todo[0], flats[0])
-        self._f_vector_rec(f_vec, flats, todo, 0, 0)
+        self._whitney_numbers2_rec(f_vec, flats, todo, 0, 0)
         for i in range(self.full_rank() + 1):
             bitset_free(flats[i])
             bitset_free(todo[i])
         sig_free(flats)
         sig_free(todo)
         return f_vec
 
-    cdef _f_vector_rec(self, object f_vec, bitset_t* flats, bitset_t* todo, long elt, long i) noexcept:
+    cdef _whitney_numbers2_rec(self, object f_vec, bitset_t* flats, bitset_t* todo, long elt, long i) noexcept:
         """
-        Recursion for the f_vector method.
+        Recursion for the whitney_numbers2 method.
         """
         cdef long e
         f_vec[i] += 1
@@ -1287,7 +1286,7 @@ cdef class BasisExchangeMatroid(Matroid):
             bitset_difference(todo[i + 1], flats[i + 1], flats[i])
             if bitset_first(todo[i + 1]) == e:
                 bitset_copy(todo[i + 1], todo[i])
-                self._f_vector_rec(f_vec, flats, todo, e + 1, i + 1)
+                self._whitney_numbers2_rec(f_vec, flats, todo, e + 1, i + 1)
             e = bitset_next(todo[i], e)
 
     cpdef flats(self, r) noexcept:
@@ -1311,7 +1310,7 @@ cdef class BasisExchangeMatroid(Matroid):
         EXAMPLES::
 
             sage: M = matroids.catalog.S8()
-            sage: M.f_vector()
+            sage: M.whitney_numbers2()
             [1, 8, 22, 14, 1]
             sage: len(M.flats(2))
             22
@@ -1386,7 +1385,7 @@ cdef class BasisExchangeMatroid(Matroid):
         EXAMPLES::
 
             sage: M = matroids.catalog.S8().dual()
-            sage: M.f_vector()
+            sage: M.whitney_numbers2()
             [1, 8, 22, 14, 1]
             sage: len(M.coflats(2))
             22
@@ -1950,15 +1949,16 @@ cdef class BasisExchangeMatroid(Matroid):
             sage: M._weak_invariant() == N._weak_invariant()
             False
         """
+        from sage.matroids.utilities import cmp_elements_key
         if self._weak_invariant_var is None:
             if self.full_rank() == 0 or self.full_corank() == 0:
                 self._weak_invariant_var = 0
                 self._weak_partition_var = SetSystem(self._E, [self.groundset()])
             else:
                 k = min(self.full_rank() - 1, 2)
                 fie, f_vec = self._flat_element_inv(k)
-                self._weak_invariant_var = hash(tuple([tuple([(f, len(fie[f])) for f in sorted(fie)]), f_vec]))
-                self._weak_partition_var = SetSystem(self._E, [fie[f] for f in sorted(fie)])
+                self._weak_invariant_var = hash(tuple([tuple([(f, len(fie[f])) for f in sorted(fie, key=cmp_elements_key)]), f_vec]))
+                self._weak_partition_var = SetSystem(self._E, [fie[f] for f in sorted(fie, key=cmp_elements_key)])
         return self._weak_invariant_var
 
     cpdef _weak_partition(self) noexcept:

src/sage/matroids/basis_matroid.pyx

@@ -428,9 +428,9 @@ cdef class BasisMatroid(BasisExchangeMatroid):
             sage: M = Matroid(bases=matroids.catalog.N2().bases())
             sage: M.truncation()
             Matroid of rank 5 on 12 elements with 702 bases
-            sage: M.f_vector()
+            sage: M.whitney_numbers2()
             [1, 12, 66, 190, 258, 99, 1]
-            sage: M.truncation().f_vector()
+            sage: M.truncation().whitney_numbers2()
             [1, 12, 66, 190, 258, 1]
         """
         if self.full_rank() == 0:

src/sage/matroids/circuits_matroid.pyx

@@ -419,7 +419,7 @@ cdef class CircuitsMatroid(Matroid):
         r"""
         Return the bases of the matroid.
 
-        OUTPUT: a SetSystem
+        OUTPUT: a :class:`SetSystem`
 
         EXAMPLES::
 
@@ -478,7 +478,7 @@ cdef class CircuitsMatroid(Matroid):
 
         - ``k`` -- an integer (optional); the length of the circuits
 
-        OUTPUT: a SetSystem
+        OUTPUT: a :class:`SetSystem`
 
         EXAMPLES::
 
@@ -541,7 +541,7 @@ cdef class CircuitsMatroid(Matroid):
         """
         Return the nonspanning circuits of the matroid.
 
-        OUTPUT: a SetSystem
+        OUTPUT: a :class:`SetSystem`
 
         EXAMPLES::
 
@@ -612,8 +612,9 @@ cdef class CircuitsMatroid(Matroid):
             sage: C1 == C2
             True
         """
+        from sage.matroids.utilities import cmp_elements_key
         if ordering is None:
-            ordering = sorted(self.groundset(), key=str)
+            ordering = sorted(self.groundset(), key=cmp_elements_key)
         else:
             if frozenset(ordering) != self.groundset():
                 raise ValueError("not an ordering of the groundset")

src/sage/matroids/database_matroids.py

@@ -541,7 +541,7 @@ def P7():
 
         sage: M = matroids.catalog.P7(); M
         P7: Ternary matroid of rank 3 on 7 elements, type 1+
-        sage: M.f_vector()
+        sage: M.whitney_numbers2()
         [1, 7, 11, 1]
         sage: M.has_minor(matroids.CompleteGraphic(4))
         False

src/sage/matroids/graphic_matroid.py

@@ -738,7 +738,7 @@ def _closure(self, X):
 
         - ``X`` -- an iterable container of ground set elements
 
-        OUTPUT: ``frozenset`` instance containing a subset of the groundset
+        OUTPUT: a subset of the groundset as a :class:`frozenset`
 
         EXAMPLES::
 
@@ -789,7 +789,7 @@ def _max_independent(self, X):
         - ``X`` -- An object with Python's ``frozenset`` interface containing
           a subset of ``self.groundset()``
 
-        OUTPUT: ``frozenset`` instance containing a subset of the groundset
+        OUTPUT: a subset of the groundset as a :class:`frozenset`
 
         EXAMPLES::
 
@@ -828,7 +828,7 @@ def _max_coindependent(self, X):
 
         - ``X`` -- an iterable container of ground set elements
 
-        OUTPUT: ``frozenset`` instance containing a subset of the groundset
+        OUTPUT: a subset of the groundset as a :class:`frozenset`
 
         EXAMPLES::
 
@@ -942,7 +942,7 @@ def _coclosure(self, X):
 
         - ``X`` -- an iterable container of ground set elements
 
-        OUTPUT: ``frozenset`` instance containing a subset of the groundset
+        OUTPUT: a subset of the groundset as a :class:`frozenset`
 
         EXAMPLES::
 

src/sage/matroids/matroid.pxd

@@ -124,9 +124,15 @@ cdef class Matroid(SageObject):
     cpdef coflats(self, r) noexcept
     cpdef hyperplanes(self) noexcept
     cpdef f_vector(self) noexcept
+    cpdef whitney_numbers(self) noexcept
+    cpdef whitney_numbers2(self) noexcept
     cpdef broken_circuits(self, ordering=*) noexcept
     cpdef no_broken_circuits_sets(self, ordering=*) noexcept
 
+    # polytopes
+    cpdef matroid_polytope(self) noexcept
+    cpdef independence_matroid_polytope(self) noexcept
+
     # isomorphism
     cpdef is_isomorphic(self, other, certificate=*) noexcept
     cpdef _is_isomorphic(self, other, certificate=*) noexcept
@@ -153,6 +159,7 @@ cdef class Matroid(SageObject):
     cpdef modular_cut(self, subsets) noexcept
     cpdef linear_subclasses(self, line_length=*, subsets=*) noexcept
     cpdef extensions(self, element=*, line_length=*, subsets=*) noexcept
+    cpdef coextensions(self, element=*, coline_length=*, subsets=*) noexcept
 
     # connectivity
     cpdef simplify(self) noexcept
@@ -213,14 +220,19 @@ cdef class Matroid(SageObject):
     cpdef _internal(self, B) noexcept
     cpdef _external(self, B) noexcept
     cpdef tutte_polynomial(self, x=*, y=*) noexcept
+    cpdef characteristic_polynomial(self, la=*) noexcept
     cpdef flat_cover(self, solver=*, verbose=*, integrality_tolerance=*) noexcept
 
     # misc
     cpdef automorphism_group(self) noexcept
     cpdef bergman_complex(self) noexcept
     cpdef augmented_bergman_complex(self) noexcept
+    cpdef broken_circuit_complex(self, ordering=*) noexcept
 
     # visualization
     cpdef plot(self,B=*,lineorders=*,pos_method=*,pos_dict=*,save_pos=*) noexcept
     cpdef show(self,B=*,lineorders=*,pos_method=*,pos_dict=*,save_pos=*,lims=*) noexcept
     cpdef _fix_positions(self,pos_dict=*,lineorders=*) noexcept
+
+    # construction
+    cpdef direct_sum(self, matroids) noexcept