Python DiGraph.set_latex_options方法代码示例

# 需要导入模块: from sage.graphs.all import DiGraph [as 别名]
# 或者: from sage.graphs.all.DiGraph import set_latex_options [as 别名]
        def digraph(self):
            """
            Returns the DiGraph associated to self.

            EXAMPLES::

                sage: C = Crystals().example(5)
                sage: C.digraph()
                Digraph on 6 vertices

            The edges of the crystal graph are by default colored using blue for edge 1, red for edge 2,
            and green for edge 3::

                sage: C = Crystals().example(3)
                sage: G = C.digraph()
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            One may also overwrite the colors::

                sage: C = Crystals().example(3)
                sage: G = C.digraph()
                sage: G.set_latex_options(color_by_label = {1:"red", 2:"purple", 3:"blue"})
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            Or one may add colors to yet unspecified edges::

                sage: C = Crystals().example(4)
                sage: G = C.digraph()
                sage: C.cartan_type()._index_set_coloring[4]="purple"
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            TODO: add more tests
            """
            from sage.graphs.all import DiGraph
            d = {}
            for x in self:
                d[x] = {}
                for i in self.index_set():
                    child = x.f(i)
                    if child is None:
                        continue
                    d[x][child]=i
            G = DiGraph(d)
            if have_dot2tex():
                G.set_latex_options(format="dot2tex", edge_labels = True, color_by_label = self.cartan_type()._index_set_coloring,
                                    edge_options = lambda (u,v,label): ({"backward":label ==0}))
            return G

# 需要导入模块: from sage.graphs.all import DiGraph [as 别名]
# 或者: from sage.graphs.all.DiGraph import set_latex_options [as 别名]

#.........这里部分代码省略.........

            INPUT:

            - ``X`` -- (optional) the vertex set `X` (default:
              the whole set of vertices of ``self`` of weight `0`)
            - ``index_set`` -- (optional) the index set `I`
              (default: the whole index set of ``self``); this has
              to be a subset of the index set of ``self`` (as a list
              or tuple)
            - ``directed`` -- (default: ``True``) whether to have the
              dual equivalence graph be directed, where the head of
              an edge `b - b'` is `b` and the tail is
              `b' = f_{i-1} f_i e_{i-1} e_i b`)

            .. SEEALSO::

                :meth:`sage.combinat.partition.Partition.dual_equivalence_graph`

            REFERENCES:

            .. [Assaf08] Sami Assaf. *A combinatorial realization of Schur-Weyl
               duality via crystal graphs and dual equivalence graphs*.
               FPSAC 2008, 141-152, Discrete Math. Theor. Comput. Sci. Proc.,
               AJ, Assoc. Discrete Math. Theor. Comput. Sci., (2008).
               :arxiv:`0804.1587v1`

            EXAMPLES::

                sage: T = crystals.Tableaux(['A',3], shape=[2,2])
                sage: G = T.dual_equivalence_graph()
                sage: sorted(G.edges())
                [([[1, 3], [2, 4]], [[1, 2], [3, 4]], 2),
                 ([[1, 2], [3, 4]], [[1, 3], [2, 4]], 3)]
                sage: T = crystals.Tableaux(['A',4], shape=[3,2])
                sage: G = T.dual_equivalence_graph()
                sage: sorted(G.edges())
                [([[1, 3, 5], [2, 4]], [[1, 3, 4], [2, 5]], 4),
                 ([[1, 3, 5], [2, 4]], [[1, 2, 5], [3, 4]], 2),
                 ([[1, 3, 4], [2, 5]], [[1, 2, 4], [3, 5]], 2),
                 ([[1, 2, 5], [3, 4]], [[1, 3, 5], [2, 4]], 3),
                 ([[1, 2, 4], [3, 5]], [[1, 2, 3], [4, 5]], 3),
                 ([[1, 2, 3], [4, 5]], [[1, 2, 4], [3, 5]], 4)]

                sage: T = crystals.Tableaux(['A',4], shape=[3,1])
                sage: G = T.dual_equivalence_graph(index_set=[1,2,3])
                sage: G.vertices()
                [[[1, 3, 4], [2]], [[1, 2, 4], [3]], [[1, 2, 3], [4]]]
                sage: G.edges()
                [([[1, 3, 4], [2]], [[1, 2, 4], [3]], 2),
                 ([[1, 2, 4], [3]], [[1, 2, 3], [4]], 3)]

            TESTS::

                sage: T = crystals.Tableaux(['A',4], shape=[3,1])
                sage: G = T.dual_equivalence_graph(index_set=[2,3])
                sage: sorted(G.edges())
                [([[1, 2, 4], [3]], [[1, 2, 3], [4]], 3),
                 ([[2, 4, 5], [3]], [[2, 3, 5], [4]], 3)]
                sage: sorted(G.vertices())
                [[[1, 3, 4], [2]],
                 [[1, 2, 4], [3]],
                 [[2, 4, 5], [3]],
                 [[1, 2, 3], [4]],
                 [[2, 3, 5], [4]],
                 [[1, 1, 1], [5]],
                 [[1, 1, 5], [5]],
                 [[1, 5, 5], [5]],
                 [[2, 3, 4], [5]]]
            """
            if index_set is None:
                index_set = self.index_set()

            def wt_zero(x):
                for i in index_set:
                    if x.epsilon(i) != x.phi(i):
                        return False
                return True

            if X is None:
                X = [x for x in self if wt_zero(x)]
                checker = lambda x: True
            elif any(not wt_zero(x) for x in X):
                raise ValueError("the elements are not all weight 0")
            else:
                checker = lambda x: x in X

            edges = []
            for x in X:
                for k, i in enumerate(index_set[1:]):
                    im = index_set[k]
                    if x.epsilon(i) == 1 and x.epsilon(im) == 0:
                        y = x.e(i).e(im).f(i).f(im)
                        if checker(y):
                            edges.append([x, y, i])
            from sage.graphs.all import DiGraph
            G = DiGraph([X, edges], format="vertices_and_edges", immutable=True)
            if have_dot2tex():
                G.set_latex_options(format="dot2tex", edge_labels=True,
                                    color_by_label=self.cartan_type()._index_set_coloring)
            return G

# 需要导入模块: from sage.graphs.all import DiGraph [as 别名]
# 或者: from sage.graphs.all.DiGraph import set_latex_options [as 别名]
    def markov_chain_digraph(self, action = 'promotion', labeling = 'identity'):
        r"""
        Returns the digraph of the action of generalized promotion or tau on ``self``

        INPUT:

        - ``action`` -- 'promotion' or 'tau' (default: 'promotion')
        - ``labeling`` -- 'identity' or 'source' (default: 'identity')

        .. todo::

            - generalize this feature by accepting a family of operators as input
            - move up in some appropriate category

        This method creates a graph with vertices being the linear extensions of a given finite
        poset and an edge from `\pi` to `\pi'` if `\pi' = \pi \partial_i` where `\partial_i` is
        the promotion operator (see :meth:`promotion`) if ``action`` is set to ``promotion``
        and `\tau_i` (see :meth:`tau`) if ``action`` is set to ``tau``. The label of the edge
        is `i` (resp. `\pi_i`) if ``labeling`` is set to ``identity`` (resp. ``source``).

        EXAMPLES::

            sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension = True)
            sage: L = P.linear_extensions()
            sage: G = L.markov_chain_digraph(); G
            Looped multi-digraph on 5 vertices
            sage: sorted(G.vertices(), key = repr)
            [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3]]
            sage: sorted(G.edges(), key = repr)
            [([1, 2, 3, 4], [1, 2, 3, 4], 4), ([1, 2, 3, 4], [1, 2, 4, 3], 2), ([1, 2, 3, 4], [1, 2, 4, 3], 3),
            ([1, 2, 3, 4], [2, 1, 4, 3], 1), ([1, 2, 4, 3], [1, 2, 3, 4], 3), ([1, 2, 4, 3], [1, 2, 4, 3], 4),
            ([1, 2, 4, 3], [1, 4, 2, 3], 2), ([1, 2, 4, 3], [2, 1, 3, 4], 1), ([1, 4, 2, 3], [1, 2, 3, 4], 1),
            ([1, 4, 2, 3], [1, 2, 3, 4], 2), ([1, 4, 2, 3], [1, 4, 2, 3], 3), ([1, 4, 2, 3], [1, 4, 2, 3], 4),
            ([2, 1, 3, 4], [1, 2, 4, 3], 1), ([2, 1, 3, 4], [2, 1, 3, 4], 4), ([2, 1, 3, 4], [2, 1, 4, 3], 2),
            ([2, 1, 3, 4], [2, 1, 4, 3], 3), ([2, 1, 4, 3], [1, 4, 2, 3], 1), ([2, 1, 4, 3], [2, 1, 3, 4], 2),
            ([2, 1, 4, 3], [2, 1, 3, 4], 3), ([2, 1, 4, 3], [2, 1, 4, 3], 4)]

            sage: G = L.markov_chain_digraph(labeling = 'source')
            sage: sorted(G.vertices(), key = repr)
            [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3]]
            sage: sorted(G.edges(), key = repr)
            [([1, 2, 3, 4], [1, 2, 3, 4], 4), ([1, 2, 3, 4], [1, 2, 4, 3], 2), ([1, 2, 3, 4], [1, 2, 4, 3], 3),
            ([1, 2, 3, 4], [2, 1, 4, 3], 1), ([1, 2, 4, 3], [1, 2, 3, 4], 4), ([1, 2, 4, 3], [1, 2, 4, 3], 3),
            ([1, 2, 4, 3], [1, 4, 2, 3], 2), ([1, 2, 4, 3], [2, 1, 3, 4], 1), ([1, 4, 2, 3], [1, 2, 3, 4], 1),
            ([1, 4, 2, 3], [1, 2, 3, 4], 4), ([1, 4, 2, 3], [1, 4, 2, 3], 2), ([1, 4, 2, 3], [1, 4, 2, 3], 3),
            ([2, 1, 3, 4], [1, 2, 4, 3], 2), ([2, 1, 3, 4], [2, 1, 3, 4], 4), ([2, 1, 3, 4], [2, 1, 4, 3], 1),
            ([2, 1, 3, 4], [2, 1, 4, 3], 3), ([2, 1, 4, 3], [1, 4, 2, 3], 2), ([2, 1, 4, 3], [2, 1, 3, 4], 1),
            ([2, 1, 4, 3], [2, 1, 3, 4], 4), ([2, 1, 4, 3], [2, 1, 4, 3], 3)]

        The edges of the graph are by default colored using blue for
        edge 1, red for edge 2, green for edge 3, and yellow for edge 4::

            sage: view(G) #optional - dot2tex graphviz

        Alternatively, one may get the graph of the action of the ``tau`` operator::

            sage: G = L.markov_chain_digraph(action='tau'); G
            Looped multi-digraph on 5 vertices
            sage: sorted(G.vertices(), key = repr)
            [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3]]
            sage: sorted(G.edges(), key = repr)
            [([1, 2, 3, 4], [1, 2, 3, 4], 2), ([1, 2, 3, 4], [1, 2, 4, 3], 3), ([1, 2, 3, 4], [2, 1, 3, 4], 1),
            ([1, 2, 4, 3], [1, 2, 3, 4], 3), ([1, 2, 4, 3], [1, 4, 2, 3], 2), ([1, 2, 4, 3], [2, 1, 4, 3], 1),
            ([1, 4, 2, 3], [1, 2, 4, 3], 2), ([1, 4, 2, 3], [1, 4, 2, 3], 1), ([1, 4, 2, 3], [1, 4, 2, 3], 3),
            ([2, 1, 3, 4], [1, 2, 3, 4], 1), ([2, 1, 3, 4], [2, 1, 3, 4], 2), ([2, 1, 3, 4], [2, 1, 4, 3], 3),
            ([2, 1, 4, 3], [1, 2, 4, 3], 1), ([2, 1, 4, 3], [2, 1, 3, 4], 3), ([2, 1, 4, 3], [2, 1, 4, 3], 2)]
            sage: view(G) #optional - dot2tex graphviz

        .. seealso:: :meth:`markov_chain_transition_matrix`, :meth:`promotion`, :meth:`tau`

        TESTS::

            sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension = True, facade = True)
            sage: L = P.linear_extensions()
            sage: G = L.markov_chain_digraph(labeling = 'source'); G
            Looped multi-digraph on 5 vertices
        """
        d = dict([x,dict([y,[]] for y in self)] for x in self)
        if action == 'promotion':
            R = range(self.poset().cardinality())
        else:
            R = range(self.poset().cardinality()-1)
        if labeling == 'source':
            for x in self:
                for i in R:
                    child = getattr(x, action)(i+1)
                    d[x][child]+=[self.poset().unwrap(x[i])]
        else:
            for x in self:
                for i in R:
                    child = getattr(x, action)(i+1)
                    d[x][child]+=[i+1]
        G = DiGraph(d)
        if have_dot2tex():
            G.set_latex_options(format="dot2tex", edge_labels = True, color_by_label = {1:"blue", 2:"red", 3:"green", 4:"yellow"})
            #G.set_latex_options(format="dot2tex", edge_labels = True, color_by_label = {1:"green", 2:"blue", 3:"brown", 4:"red"})
        return G

# 需要导入模块: from sage.graphs.all import DiGraph [as 别名]
# 或者: from sage.graphs.all.DiGraph import set_latex_options [as 别名]
        def digraph(self, subset=None, index_set=None):
            """
            Returns the DiGraph associated to ``self``.

            INPUT:

            - ``subset`` -- (Optional) A subset of vertices for
              which the digraph should be constructed

            - ``index_set`` -- (Optional) The index set to draw arrows

            EXAMPLES::

                sage: C = Crystals().example(5)
                sage: C.digraph()
                Digraph on 6 vertices

            The edges of the crystal graph are by default colored using blue for edge 1, red for edge 2,
            and green for edge 3::

                sage: C = Crystals().example(3)
                sage: G = C.digraph()
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            One may also overwrite the colors::

                sage: C = Crystals().example(3)
                sage: G = C.digraph()
                sage: G.set_latex_options(color_by_label = {1:"red", 2:"purple", 3:"blue"})
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            Or one may add colors to yet unspecified edges::

                sage: C = Crystals().example(4)
                sage: G = C.digraph()
                sage: C.cartan_type()._index_set_coloring[4]="purple"
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            Here is an example of how to take the top part up to a given depth of an infinite dimensional
            crystal::

                sage: C = CartanType(['C',2,1])
                sage: La = C.root_system().weight_lattice().fundamental_weights()
                sage: T = HighestWeightCrystal(La[0])
                sage: S = T.subcrystal(max_depth=3)
                sage: G = T.digraph(subset=S); G
                Digraph on 5 vertices
                sage: G.vertices()
                [(1/2*Lambda[0] + Lambda[1] - Lambda[2] - 1/2*delta, -1/2*Lambda[0] + Lambda[1] - 1/2*delta),
                (-Lambda[0] + 2*Lambda[1] - delta,), (Lambda[0] - 2*Lambda[1] + 2*Lambda[2] - delta,),
                (1/2*Lambda[0] - Lambda[1] + Lambda[2] - 1/2*delta, -1/2*Lambda[0] + Lambda[1] - 1/2*delta), (Lambda[0],)]

            Here is a way to construct a picture of a Demazure crystal using
            the ``subset`` option::

                sage: B = CrystalOfTableaux(['A',2], shape=[2,1])
                sage: C = CombinatorialFreeModule(QQ,B)
                sage: t = B.highest_weight_vector()
                sage: b = C(t)
                sage: D = B.demazure_operator(b,[2,1]); D
                B[[[1, 1], [2]]] + B[[[1, 2], [2]]] + B[[[1, 3], [2]]] + B[[[1, 1], [3]]] + B[[[1, 3], [3]]]
                sage: G = B.digraph(subset=D.support())
                sage: G.vertices()
                [[[1, 1], [2]], [[1, 2], [2]], [[1, 3], [2]], [[1, 1], [3]], [[1, 3], [3]]]
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            We can also choose to display particular arrows using the
            ``index_set`` option::

                sage: C = KirillovReshetikhinCrystal(['D',4,1], 2, 1)
                sage: G = C.digraph(index_set=[1,3])
                sage: len(G.edges())
                20
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            TODO: add more tests
            """
            from sage.graphs.all import DiGraph
            from sage.categories.highest_weight_crystals import HighestWeightCrystals
            d = {}
            if self in HighestWeightCrystals:
                f = lambda (u,v,label): ({})
            else:
                f = lambda (u,v,label): ({"backward":label ==0})

            # Parse optional arguments
            if subset is None:
                subset = self
            if index_set is None:
                index_set = self.index_set()

            for x in subset:
                d[x] = {}
                for i in index_set:
                    child = x.f(i)
                    if child is None or child not in subset:
                        continue
                    d[x][child]=i
            G = DiGraph(d)
            if have_dot2tex():
#.........这里部分代码省略.........

# 需要导入模块: from sage.graphs.all import DiGraph [as 别名]
# 或者: from sage.graphs.all.DiGraph import set_latex_options [as 别名]
        def digraph(self, subset=None, index_set=None, depth=None):
            """
            Return the DiGraph associated to ``self``.

            INPUT:

            - ``subset`` -- (optional) a subset of vertices for
              which the digraph should be constructed

            - ``index_set`` -- (optional) the index set to draw arrows

            - ``depth`` -- the depth to draw; optional only for finite crystals

            EXAMPLES::

                sage: T = crystals.Tableaux(['A',2], shape=[2,1])
                sage: T.digraph()
                Digraph on 8 vertices
                sage: S = T.subcrystal(max_depth=2)
                sage: len(S)
                5
                sage: G = T.digraph(subset=list(S))
                sage: G.is_isomorphic(T.digraph(depth=2), edge_labels=True)
                True

            TESTS:

            The following example demonstrates the speed improvement.
            The speedup in non-affine types is small however::

                sage: depth = 5
                sage: C = crystals.AlcovePaths(['A',2,1], [1,1,0])
                sage: general_digraph = Crystals().parent_class.digraph
                sage: S = C.subcrystal(max_depth=depth, direction='lower')
                sage: %timeit C.digraph(depth=depth) # not tested
                10 loops, best of 3: 48.9 ms per loop
                sage: %timeit general_digraph(C, subset=S) # not tested
                10 loops, best of 3: 96.5 ms per loop
                sage: G1 = C.digraph(depth=depth)
                sage: G2 = general_digraph(C, subset=S)
                sage: G1.is_isomorphic(G2, edge_labels=True)
                True
            """
            if subset is not None:
                return Crystals().parent_class.digraph(self, subset, index_set)

            if self not in Crystals().Finite() and depth is None:
                raise NotImplementedError("crystals not known to be finite must"
                                          " specify either the subset or depth")

            from sage.graphs.all import DiGraph
            if index_set is None:
                index_set = self.index_set()

            rank = 0
            d = {g: {} for g in self.module_generators}
            visited = set(d.keys())

            while depth is None or rank < depth:
                recently_visited = set()
                for x in visited:
                    d.setdefault(x, {}) # does nothing if there's a default
                    for i in index_set:
                        xfi = x.f(i)
                        if xfi is not None:
                            d[x][xfi] = i
                            recently_visited.add(xfi)
                if not recently_visited: # No new nodes, nothing more to do
                    break
                rank += 1
                visited = recently_visited

            G = DiGraph(d)
            if have_dot2tex():
                G.set_latex_options(format="dot2tex",
                                    edge_labels=True,
                                    color_by_label=self.cartan_type()._index_set_coloring)
            return G