def to_non_crossing_set_partition(self):
        r"""
        Returns the noncrossing set partition (on half as many elements) 
        corresponding to the perfect matching if the perfect matching is 
        noncrossing, and otherwise gives an error.

        OUTPUT:

            The realization of ``self`` as a noncrossing set partition.

        EXAMPLES::

            sage: PerfectMatching([[1,3], [4,2]]).to_non_crossing_set_partition()
            Traceback (most recent call last):
            ...
            ValueError: matching must be non-crossing
            sage: PerfectMatching([[1,4], [3,2]]).to_non_crossing_set_partition()
            {{1, 2}}
            sage: PerfectMatching([]).to_non_crossing_set_partition()
            {}
        """
        from sage.combinat.set_partition import SetPartition        
        if not self.is_non_crossing():
            raise ValueError("matching must be non-crossing")
        else:
            perm = self.to_permutation()
            perm2 = Permutation([(perm[2*i])/2 for i in range(len(perm)/2)])
        return SetPartition(perm2.cycle_tuples())

    def verify_representation(self):
        r"""
        Verify the representation: tests that the images of the simple
        transpositions are involutions and tests that the braid relations
        hold.

        EXAMPLES::

            sage: spc = SymmetricGroupRepresentation([1,1,1])
            sage: spc.verify_representation()
            True
            sage: spc = SymmetricGroupRepresentation([4,2,1])
            sage: spc.verify_representation()
            True
        """
        n = self._partition.size()
        transpositions = []
        for i in range(1,n):
            si = Permutation(range(1,i) + [i+1,i] + range(i+2,n+1))
            transpositions.append(si)
        repn_matrices = map(self.representation_matrix, transpositions)
        for (i,si) in enumerate(repn_matrices):
            for (j,sj) in enumerate(repn_matrices):
                if i == j:
                    if si*sj != si.parent().identity_matrix():
                        return False, "si si != 1 for i = %s" % (i,)
                elif abs(i-j) > 1:
                    if si*sj != sj*si:
                        return False, "si sj != sj si for (i,j) =(%s,%s)" % (i,j)
                else:
                    if si*sj*si != sj*si*sj:
                        return False, "si sj si != sj si sj for (i,j) = (%s,%s)" % (i,j)
        return True

        def _element_constructor_(self, x):
            r"""
            Convert ``x`` into ``self``.

            EXAMPLES::

                sage: R = algebras.FQSym(QQ).G()
                sage: x, y, z = R([1]), R([2,1]), R([3,2,1])
                sage: R(x)
                G[1]
                sage: R(x+4*y)
                G[1] + 4*G[2, 1]
                sage: R(1)
                G[]

                sage: D = algebras.FQSym(ZZ).G()
                sage: X, Y, Z = D([1]), D([2,1]), D([3,2,1])
                sage: R(X-Y).parent()
                Free Quasi-symmetric functions over Rational Field in the G basis

                sage: R([1, 3, 2])
                G[1, 3, 2]
                sage: R(Permutation([1, 3, 2]))
                G[1, 3, 2]
                sage: R(SymmetricGroup(4)(Permutation([1,3,4,2])))
                G[1, 3, 4, 2]

                sage: RF = algebras.FQSym(QQ).F()
                sage: R(RF([2, 3, 4, 1]))
                G[4, 1, 2, 3]
                sage: R(RF([3, 2, 4, 1]))
                G[4, 2, 1, 3]
                sage: DF = algebras.FQSym(ZZ).F()
                sage: D(DF([2, 3, 4, 1]))
                G[4, 1, 2, 3]
                sage: R(DF([2, 3, 4, 1]))
                G[4, 1, 2, 3]
                sage: RF(R[2, 3, 4, 1])
                F[4, 1, 2, 3]
            """
            if isinstance(x, (list, tuple, PermutationGroupElement)):
                x = Permutation(x)
            try:
                P = x.parent()
                if isinstance(P, FreeQuasisymmetricFunctions.G):
                    if P is self:
                        return x
                    return self.element_class(self, x.monomial_coefficients())
            except AttributeError:
                pass
            return CombinatorialFreeModule._element_constructor_(self, x)

    def SymmetricGroupBruhatIntervalPoset(start, end):
        """
        The poset of permutations with respect to Bruhat order.

        INPUT:

        - ``start`` - list permutation

        - ``end`` - list permutation (same n, of course)

        .. note::

           Must have ``start`` <= ``end``.

        EXAMPLES:

        Any interval is rank symmetric if and only if it avoids these
        permutations::

            sage: P1 = Posets.SymmetricGroupBruhatIntervalPoset([1,2,3,4], [3,4,1,2])
            sage: P2 = Posets.SymmetricGroupBruhatIntervalPoset([1,2,3,4], [4,2,3,1])
            sage: ranks1 = [P1.rank(v) for v in P1]
            sage: ranks2 = [P2.rank(v) for v in P2]
            sage: [ranks1.count(i) for i in uniq(ranks1)]
            [1, 3, 5, 4, 1]
            sage: [ranks2.count(i) for i in uniq(ranks2)]
            [1, 3, 5, 6, 4, 1]

        """
        start = Permutation(start)
        end = Permutation(end)
        if len(start) != len(end):
            raise TypeError("Start (%s) and end (%s) must have same length."%(start, end))
        if not start.bruhat_lequal(end):
            raise TypeError("Must have start (%s) <= end (%s) in Bruhat order."%(start, end))
        unseen = [start]
        nodes = {}
        while len(unseen) > 0:
            perm = unseen.pop(0)
            nodes[perm] = [succ_perm for succ_perm in perm.bruhat_succ()
                                if succ_perm.bruhat_lequal(end)]
            for succ_perm in nodes[perm]:
                if succ_perm not in nodes:
                    unseen.append(succ_perm)
        return Poset(nodes)

    def scalar_product_matrix(self, permutation=None):
        r"""
        Return the scalar product matrix corresponding to ``permutation``.

        The entries are given by the scalar products of ``u`` and
        ``permutation.action(v)``, where ``u`` is a vertex in the underlying
        Yang-Baxter graph and ``v`` is a vertex in the dual graph.

        EXAMPLES::

            sage: spc = SymmetricGroupRepresentation([3,1], 'specht')
            sage: spc.scalar_product_matrix()
            [ 1  0  0]
            [ 0 -1  0]
            [ 0  0  1]
        """
        if permutation is None:
            permutation = Permutation(range(1,1+self._partition.size()))
        Q = matrix(QQ, len(self._yang_baxter_graph))
        for (i,v) in enumerate(self._dual_vertices):
            for (j,u) in enumerate(self._yang_baxter_graph):
                Q[i,j] = self.scalar_product(tuple(permutation.action(v)), u)
        return Q

def PermutationGraph(second_permutation, first_permutation = None):
    r"""
    Build a permutation graph from one permutation or from two lists.

    Definition:

    If `\sigma` is a permutation of `\{ 1, 2, \ldots, n \}`, then the
    permutation graph of `\sigma` is the graph on vertex set
    `\{ 1, 2, \ldots, n \}` in which two vertices `i` and `j` satisfying
    `i < j` are connected by an edge if and only if
    `\sigma^{-1}(i) > \sigma^{-1}(j)`. A visual way to construct this
    graph is as follows:

      Take two horizontal lines in the euclidean plane, and mark points
      `1, ..., n` from left to right on the first of them. On the second
      one, still from left to right, mark `n` points
      `\sigma(1), \sigma(2), \ldots, \sigma(n)`.
      Now, link by a segment the two points marked with `1`, then link
      together the points marked with `2`, and so on. The permutation
      graph of `\sigma` is the intersection graph of those segments: there
      exists a vertex in this graph for each element from `1` to `n`, two
      vertices `i, j` being adjacent if the segments `i` and `j` cross
      each other.

    The set of edges of the permutation graph can thus be identified with
    the set of inversions of the inverse of the given permutation
    `\sigma`.

    A more general notion of permutation graph can be defined as
    follows: If `S` is a set, and `(a_1, a_2, \ldots, a_n)` and
    `(b_1, b_2, \ldots, b_n)` are two lists of elements of `S`, each of
    which lists contains every element of `S` exactly once, then the
    permutation graph defined by these two lists is the graph on the
    vertex set `S` in which two vertices `i` and `j` are connected by an
    edge if and only if the order in which these vertices appear in the
    list `(a_1, a_2, \ldots, a_n)` is the opposite of the order in which
    they appear in the list `(b_1, b_2, \ldots, b_n)`. When
    `(a_1, a_2, \ldots, a_n) = (1, 2, \ldots, n)`, this graph is the
    permutation graph of the permutation
    `(b_1, b_2, \ldots, b_n) \in S_n`. Notice that `S` does not have to
    be a set of integers here, but can be a set of strings, tuples, or
    anything else. We can still use the above visual description to
    construct the permutation graph, but now we have to mark points
    `a_1, a_2, \ldots, a_n` from left to right on the first horizontal
    line and points `b_1, b_2, \ldots, b_n` from left to right on the
    second horizontal line.

    INPUT:

    - ``second_permutation`` -- the unique permutation/list defining the graph,
      or the second of the two (if the graph is to be built from two
      permutations/lists).

    - ``first_permutation`` (optional) -- the first of the two
      permutations/lists from which the graph should be built, if it is to be
      built from two permutations/lists.

      When ``first_permutation is None`` (default), it is set to be equal to
      ``sorted(second_permutation)``, which yields the expected ordering when
      the elements of the graph are integers.

    .. SEEALSO:

      - Recognition of Permutation graphs in the :mod:`comparability module
        <sage.graphs.comparability>`.

      - Drawings of permutation graphs as intersection graphs of segments is
        possible through the
        :meth:`~sage.combinat.permutation.Permutation.show` method of
        :class:`~sage.combinat.permutation.Permutation` objects.

        The correct argument to use in this case is ``show(representation =
        "braid")``.

      - :meth:`~sage.combinat.permutation.Permutation.inversions`

    EXAMPLES::

        sage: p = Permutations(5).random_element()
        sage: PG = graphs.PermutationGraph(p)
        sage: edges = PG.edges(labels=False)
        sage: set(edges) == set(p.inverse().inversions())
        True

        sage: PG = graphs.PermutationGraph([3,4,5,1,2])
        sage: sorted(PG.edges())
        [(1, 3, None),
         (1, 4, None),
         (1, 5, None),
         (2, 3, None),
         (2, 4, None),
         (2, 5, None)]
        sage: PG = graphs.PermutationGraph([3,4,5,1,2], [1,4,2,5,3])
        sage: sorted(PG.edges())
        [(1, 3, None),
         (1, 4, None),
         (1, 5, None),
         (2, 3, None),
         (2, 5, None),
         (3, 4, None),
#.........这里部分代码省略.........

    def __classcall_private__(cls, p):
        r"""
        This function tries to recognize the input (it can be either a list or
        a tuple of pairs, or a fix-point free involution given as a list or as
        a permutation), constructs the parent (enumerated set of
        PerfectMatchings of the ground set) and calls the __init__ function to
        construct our object.

        EXAMPLES::

            sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')]);m
            [('a', 'e'), ('b', 'c'), ('d', 'f')]
            sage: isinstance(m,PerfectMatching)
            True
            sage: n = PerfectMatching([3, 8, 1, 7, 6, 5, 4, 2]);n
            [(1, 3), (2, 8), (4, 7), (5, 6)]
            sage: n.parent()
            Set of perfect matchings of {1, 2, 3, 4, 5, 6, 7, 8}
            sage: PerfectMatching([(1, 4), (2, 3), (5, 6)]).is_non_crossing()
            True

        The function checks that the given list or permutation is a valid perfect
        matching (i.e. a list of pairs with pairwise disjoint elements  or a
        fixpoint-free involution) and raises a ValueError otherwise:

            sage: PerfectMatching([(1, 2, 3), (4, 5)])
            Traceback (most recent call last):
            ...
            ValueError: [(1, 2, 3), (4, 5)] is not a valid perfect matching: all elements of the list must be pairs

        If you know your datas are in a good format, use directly
        ``PerfectMatchings(objects)(data)``.

        TESTS::

             sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')])
             sage: TestSuite(m).run()
             sage: m = PerfectMatching([])
             sage: TestSuite(m).run()
             sage: PerfectMatching(6)
             Traceback (most recent call last):
             ...
             ValueError: cannot convert p (= 6) to a PerfectMatching
             sage: PerfectMatching([(1,2,3)])
             Traceback (most recent call last):
             ...
             ValueError: [(1, 2, 3)] is not a valid perfect matching:
             all elements of the list must be pairs
             sage: PerfectMatching([(1,1)])
             Traceback (most recent call last):
             ...
             ValueError: [(1, 1)] is not a valid perfect matching:
             there are some repetitions
             sage: PerfectMatching(Permutation([4,2,1,3]))
             Traceback (most recent call last):
             ...
             ValueError: The permutation p (= [4, 2, 1, 3]) is not a fixed point free involution
        """
        # we have to extract from the argument p the set of objects of the
        # matching and the list of pairs.
        # First case: p is a list (resp tuple) of lists (resp tuple).
        if (isinstance(p, list) or isinstance(p, tuple)) and (
                all([isinstance(x, list) or isinstance(x, tuple) for x in p])):
            objects = Set(flatten(p))
            data = (map(tuple, p))
            #check if the data are correct
            if not all([len(t) == 2 for t in data]):
                raise ValueError("%s is not a valid perfect matching:\n"
                                 "all elements of the list must be pairs" % p)
            if len(objects) < 2*len(data):
                raise ValueError("%s is not a valid perfect matching:\n"
                                 "there are some repetitions" % p)
        # Second case: p is a permutation or a list of integers, we have to
        # check if it is a fix-point-free involution.
        elif ((isinstance(p, list) and
               all(map(lambda x: (isinstance(x, Integer) or isinstance(x, int)), p)))
              or isinstance(p, Permutation)):
            p = Permutation(p)
            n = len(p)
            if not(p.cycle_type() == [2 for i in range(n//2)]):
                raise ValueError("The permutation p (= %s) is not a "
                                 "fixed point free involution" % p)
            objects = Set(range(1, n+1))
            data = p.to_cycles()
        # Third case: p is already a perfect matching, we return p directly
        elif isinstance(p, PerfectMatching):
            return p
        else:
            raise ValueError("cannot convert p (= %s) to a PerfectMatching" % p)
        # Finally, we create the parent and the element using the element
        # class of the parent. Note: as this function is private, when we
        # create an object via parent.element_class(...), __init__ is directly
        # executed and we do not have an infinite loop.
        return PerfectMatchings(objects)(data)

#.........这里部分代码省略.........
        This is compatible when a permutation is given as input::

            sage: a = X([3,2,4,1])
            sage: a.divided_difference([2,3,1])
            0
            sage: a.divided_difference(1).divided_difference(2)
            0

        ::

            sage: a = X([4,3,2,1])
            sage: a.divided_difference([2,3,1])
            X[3, 2, 4, 1]
            sage: a.divided_difference(1).divided_difference(2)
            X[3, 2, 4, 1]
            sage: a.divided_difference([4,1,3,2])
            X[1, 4, 2, 3]
            sage: b = X([4, 1, 3, 2])
            sage: b.divided_difference(1).divided_difference(2)
            X[1, 3, 4, 2]
            sage: b.divided_difference(1).divided_difference(2).divided_difference(3)
            X[1, 3, 2]
            sage: b.divided_difference(1).divided_difference(2).divided_difference(3).divided_difference(2)
            X[1]
            sage: b.divided_difference(1).divided_difference(2).divided_difference(3).divided_difference(3)
            0
            sage: b.divided_difference(1).divided_difference(2).divided_difference(1)
            0

        TESTS:

        Check that :trac:`23403` is fixed::

            sage: X = SchubertPolynomialRing(ZZ)
            sage: a = X([3,2,4,1])
            sage: a.divided_difference(2)
            0
            sage: a.divided_difference([3,2,1])
            0
            sage: a.divided_difference(0)
            Traceback (most recent call last):
            ...
            ValueError: cannot apply \delta_{0} to a (= X[3, 2, 4, 1])
        """
        if not self: # if self is 0
            return self
        Perms = Permutations()
        if i in ZZ:
            if algorithm == "sage":
                if i <= 0:
                    raise ValueError(r"cannot apply \delta_{%s} to a (= %s)" % (i, self))
                # The operator `\delta_i` sends the Schubert
                # polynomial `X_\pi` (where `\pi` is a finitely supported
                # permutation of `\{1, 2, 3, \ldots\}`) to:
                # - the Schubert polynomial X_\sigma`, where `\sigma` is
                #   obtained from `\pi` by switching the values at `i` and `i+1`,
                #   if `i` is a descent of `\pi` (that is, `\pi(i) > \pi(i+1)`);
                # - `0` otherwise.
                # Notice that distinct `\pi`s lead to distinct `\sigma`s,
                # so we can use `_from_dict` here.
                res_dict = {}
                for pi, coeff in self:
                    pi = pi[:]
                    n = len(pi)
                    if n <= i:
                        continue
                    if pi[i-1] < pi[i]:
                        continue
                    pi[i-1], pi[i] = pi[i], pi[i-1]
                    pi = Perms(pi).remove_extra_fixed_points()
                    res_dict[pi] = coeff
                return self.parent()._from_dict(res_dict)
            else: # if algorithm == "symmetrica":
                return symmetrica.divdiff_schubert(i, self)
        elif i in Perms:
            if algorithm == "sage":
                i = Permutation(i)
                redw = i.reduced_word()
                res_dict = {}
                for pi, coeff in self:
                    next_pi = False
                    pi = pi[:]
                    n = len(pi)
                    for j in redw:
                        if n <= j:
                            next_pi = True
                            break
                        if pi[j-1] < pi[j]:
                            next_pi = True
                            break
                        pi[j-1], pi[j] = pi[j], pi[j-1]
                    if next_pi:
                        continue
                    pi = Perms(pi).remove_extra_fixed_points()
                    res_dict[pi] = coeff
                return self.parent()._from_dict(res_dict)
            else: # if algorithm == "symmetrica":
                return symmetrica.divdiff_perm_schubert(i, self)
        else:
            raise TypeError("i must either be an integer or permutation")

    def __classcall_private__(cls, parts):
        """
        Create a perfect matching from ``parts`` with the appropriate parent.

        This function tries to recognize the input (it can be either a list or
        a tuple of pairs, or a fix-point free involution given as a list or as
        a permutation), constructs the parent (enumerated set of
        PerfectMatchings of the ground set) and calls the __init__ function to
        construct our object.

        EXAMPLES::

            sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')]);m
            [('a', 'e'), ('b', 'c'), ('d', 'f')]
            sage: isinstance(m, PerfectMatching)
            True
            sage: n = PerfectMatching([3, 8, 1, 7, 6, 5, 4, 2]);n
            [(1, 3), (2, 8), (4, 7), (5, 6)]
            sage: n.parent()
            Perfect matchings of {1, 2, 3, 4, 5, 6, 7, 8}
            sage: PerfectMatching([(1, 4), (2, 3), (5, 6)]).is_noncrossing()
            True

        The function checks that the given list or permutation is
        a valid perfect matching (i.e. a list of pairs with pairwise
        disjoint elements or a fix point free involution) and raises
        a ``ValueError`` otherwise::

            sage: PerfectMatching([(1, 2, 3), (4, 5)])
            Traceback (most recent call last):
            ...
            ValueError: [(1, 2, 3), (4, 5)] is not an element of
             Perfect matchings of {1, 2, 3, 4, 5}

        TESTS::

            sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')])
            sage: TestSuite(m).run()
            sage: m = PerfectMatching([])
            sage: TestSuite(m).run()
            sage: PerfectMatching(6)
            Traceback (most recent call last):
            ...
            TypeError: 'sage.rings.integer.Integer' object is not iterable
            sage: PerfectMatching([(1,2,3)])
            Traceback (most recent call last):
            ...
            ValueError: [(1, 2, 3)] is not an element of
             Perfect matchings of {1, 2, 3}

            sage: PerfectMatching([(1,1)])
            Traceback (most recent call last):
            ...
            ValueError: [(1)] is not an element of Perfect matchings of {1}

            sage: PerfectMatching(Permutation([4,2,1,3]))
            Traceback (most recent call last):
            ...
            ValueError: permutation p (= [4, 2, 1, 3]) is not a
             fixed point free involution
        """
        if ((isinstance(parts, list) and
             all((isinstance(x, (int, Integer)) for x in parts)))
            or isinstance(parts, Permutation)):
            s = Permutation(parts)
            if not all(e == 2 for e in s.cycle_type()):
                raise ValueError("permutation p (= {}) is not a "
                                 "fixed point free involution".format(s))
            parts = s.to_cycles()

        base_set = frozenset(e for p in parts for e in p)
        P = PerfectMatchings(base_set)
        return P(parts)

def PermutationGraph(second_permutation, first_permutation = None):
    r"""
    Builds a permutation graph from one (or two) permutations.

    General definition

    A Permutation Graph can be encoded by a permutation `\sigma`
    of `1, ..., n`. It is then built in the following way :

      Take two horizontal lines in the euclidean plane, and mark points `1, ...,
      n` from left to right on the first of them. On the second one, still from
      left to right, mark point in the order in which they appear in `\sigma`.
      Now, link by a segment the two points marked with 1, then link together
      the points marked with 2, and so on. The permutation graph defined by the
      permutation is the intersection graph of those segments : there exists a
      point in this graph for each element from `1` to `n`, two vertices `i, j`
      being adjacent if the segments `i` and `j` cross each other.

    The set of edges of the resulting graph is equal to the set of inversions of
    the inverse of the given permutation.

    INPUT:

    - ``second_permutation`` -- the permutation from which the graph should be
      built. It corresponds to the ordering of the elements on the second line
      (see previous definition)

    - ``first_permutation`` (optional) -- the ordering of the elements on the
      *first* line. This is useful when the elements have no natural ordering,
      for instance when they are strings, or tuples, or anything else.

      When ``first_permutation == None`` (default), it is set to be equal to
      ``sorted(second_permutation)``, which just yields the expected
      ordering when the elements of the graph are integers.

    .. SEEALSO:

      - Recognition of Permutation graphs in the :mod:`comparability module
        <sage.graphs.comparability>`.

      - Drawings of permutation graphs as intersection graphs of segments is
        possible through the
        :meth:`~sage.combinat.permutation.Permutation.show` method of
        :class:`~sage.combinat.permutation.Permutation` objects.

        The correct argument to use in this case is ``show(representation =
        "braid")``.

      - :meth:`~sage.combinat.permutation.Permutation.inversions`

    EXAMPLE::

        sage: p = Permutations(5).random_element()
        sage: edges = graphs.PermutationGraph(p).edges(labels =False)
        sage: set(edges) == set(p.inverse().inversions())
        True

    TESTS::

        sage: graphs.PermutationGraph([1, 2, 3], [4, 5, 6])
        Traceback (most recent call last):
        ...
        ValueError: The two permutations do not contain the same set of elements ...
    """
    if first_permutation == None:
        first_permutation = sorted(second_permutation)
    else:
        if set(second_permutation) != set(first_permutation):
            raise ValueError("The two permutations do not contain the same "+
                             "set of elements ! It is going to be pretty "+
                             "hard to define a permutation graph from that !")

    vertex_to_index = {}
    for i, v in enumerate(first_permutation):
        vertex_to_index[v] = i+1

    from sage.combinat.permutation import Permutation
    p2 = Permutation(map(lambda x:vertex_to_index[x], second_permutation))
    p1 = Permutation(map(lambda x:vertex_to_index[x], first_permutation))
    p2 = p2 * p1.inverse()
    p2 = p2.inverse()

    g = Graph(name="Permutation graph for "+str(second_permutation))
    g.add_vertices(second_permutation)

    for u,v in p2.inversions():
        g.add_edge(first_permutation[u-1], first_permutation[v-1])

    return g