Module ImportFrom alias alias ImportFrom alias ImportFrom alias alias ImportFrom alias ImportFrom alias ImportFrom alias FunctionDef arguments args NameParam defaults body Expr Str For NameStore Call NameLoad Call NameLoad Call NameLoad NameLoad NameLoad body Expr Yield Call NameLoad NameLoad decorator_list FunctionDef arguments args NameParam defaults body Expr Str Assign NameStore Call NameLoad Call NameLoad NameLoad For NameStore Call NameLoad NameLoad body Expr Yield Call NameLoad NameLoad Assign NameStore Call NameLoad NameLoad Num decorator_list FunctionDef arguments args NameParam defaults body Expr Str For NameStore Call NameLoad Call NameLoad Call NameLoad NameLoad NameLoad body Assign NameStore Call NameLoad NameLoad If AttributeLoad NameLoad attr body Expr Yield NameLoad decorator_list FunctionDef arguments args NameParam defaults body Expr Str If CompareEq NameLoad Num body Expr Yield Call NameLoad ListLoad Num Num Expr Yield Call NameLoad ListLoad Num Num orelse If 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Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num ListLoad TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num ListLoad TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num TupleLoad Num Num Num Num Return ListComp Call NameLoad ListComp ListComp BinOpSub NameLoad Num comprehension NameStore NameLoad comprehension NameStore NameLoad keyword Num comprehension NameStore NameLoad decorator_list FunctionDef arguments args NameParam defaults body Expr Str If CompareLt NameLoad Num body Raise Call NameLoad Str FunctionDef arguments args NameParam NameParam defaults body Return Call AttributeLoad SubscriptLoad NameLoad Index NameLoad attr BinOpSub NameLoad NameLoad decorator_list FunctionDef arguments args NameParam NameParam defaults body Return Call AttributeLoad SubscriptLoad NameLoad Index NameLoad attr BinOpSub NameLoad Num decorator_list FunctionDef arguments args NameParam NameParam defaults body Return Call AttributeLoad SubscriptLoad NameLoad Index NameLoad attr BinOpSub NameLoad Num decorator_list FunctionDef arguments args NameParam NameParam defaults body Return Call AttributeLoad SubscriptLoad NameLoad Index NameLoad attr BinOpSub NameLoad NameLoad decorator_list FunctionDef arguments args NameParam NameParam NameParam defaults body Assign SubscriptStore SubscriptLoad NameLoad Index NameLoad ExtSlice Slice Index BinOpSub NameLoad NameLoad Call NameLoad NameLoad Num NameLoad decorator_list FunctionDef arguments args NameParam NameParam NameParam defaults body Assign SubscriptStore SubscriptLoad NameLoad Index NameLoad ExtSlice Slice Index BinOpSub NameLoad Num Call NameLoad NameLoad Num NameLoad decorator_list FunctionDef arguments args NameParam NameParam NameParam defaults body Assign SubscriptStore SubscriptLoad NameLoad Index NameLoad ExtSlice Index BinOpSub NameLoad Num Slice Call NameLoad Num NameLoad NameLoad decorator_list FunctionDef arguments args NameParam NameParam NameParam defaults body Assign SubscriptStore SubscriptLoad NameLoad Index NameLoad ExtSlice Index BinOpSub NameLoad NameLoad Slice Call NameLoad Num NameLoad NameLoad decorator_list FunctionDef arguments args NameParam NameParam defaults Num body For NameStore Call NameLoad NameLoad body Assign NameStore SubscriptLoad NameLoad Index NameLoad Assign NameStore ListLoad For NameStore Call NameLoad NameLoad body For NameStore Call NameLoad BinOpSub NameLoad Num Num Num body Expr Call AttributeLoad NameLoad attr SubscriptLoad NameLoad Index TupleLoad NameLoad NameLoad Assign SubscriptStore NameLoad Index NameLoad Call NameLoad NameLoad NameLoad NameLoad decorator_list FunctionDef arguments args NameParam defaults body Expr Call NameLoad NameLoad Num decorator_list FunctionDef arguments args NameParam NameParam defaults Num body For NameStore Call NameLoad NameLoad body If 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NameStore NameStore NameStore NameStore Call NameLoad Str Assign NameStore Dict Assign NameStore Num For NameStore Call NameLoad Num body Assign NameStore ListLoad For NameStore Call NameLoad BinOpPow NameLoad Num body Expr Call AttributeLoad NameLoad attr NameLoad AugAssignAdd NameStore Num Assign SubscriptStore NameLoad Index SubscriptLoad NameLoad Index NameLoad Call NameLoad NameLoad NameLoad NameLoad FunctionDef arguments args NameParam defaults Num body Assign NameStore ListLoad For NameStore NameLoad body Expr Call AttributeLoad NameLoad attr SubscriptLoad NameLoad Index NameLoad If NameLoad body Return NameLoad Expr Call AttributeLoad NameLoad attr Call NameLoad NameLoad decorator_list Assign NameStore ListLoad Assign NameStore Call NameLoad Call NameLoad BinOpMult Num BinOpPow NameLoad Num For NameStore Call NameLoad BinOpSub NameLoad Num body Expr Call NameLoad NameLoad Expr Call NameLoad Expr Call NameLoad NameLoad Assert CompareEq Call NameLoad Num NameLoad Expr Call 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N/A __future__ print_function division sympy.combinatorics.permutations Permutation sympy.utilities.iterables variations rotate_left sympy.core.symbol symbols sympy.matrices Matrix sympy.core.compatibility range symmetric N/A N/A n N/A N/A N/A 
    Generates the symmetric group of order n, Sn.

    Examples
    ========

    >>> from sympy.combinatorics.permutations import Permutation
    >>> Permutation.print_cyclic = True
    >>> from sympy.combinatorics.generators import symmetric
    >>> list(symmetric(3))
    [(2), (1 2), (2)(0 1), (0 1 2), (0 2 1), (0 2)]
     N/A perm N/A variations N/A list N/A range n n N/A N/A N/A N/A Permutation perm N/A cyclic N/A N/A n N/A N/A N/A 
    Generates the cyclic group of order n, Cn.

    Examples
    ========

    >>> from sympy.combinatorics.permutations import Permutation
    >>> Permutation.print_cyclic = True
    >>> from sympy.combinatorics.generators import cyclic
    >>> list(cyclic(5))
    [(4), (0 1 2 3 4), (0 2 4 1 3),
     (0 3 1 4 2), (0 4 3 2 1)]

    See Also
    ========
    dihedral
     N/A gen N/A list N/A range n N/A i N/A range n N/A N/A N/A N/A Permutation gen N/A gen N/A rotate_left gen 1 N/A alternating N/A N/A n N/A N/A N/A 
    Generates the alternating group of order n, An.

    Examples
    ========

    >>> from sympy.combinatorics.permutations import Permutation
    >>> Permutation.print_cyclic = True
    >>> from sympy.combinatorics.generators import alternating
    >>> list(alternating(3))
    [(2), (0 1 2), (0 2 1)]
     N/A perm N/A variations N/A list N/A range n n N/A N/A p N/A Permutation perm N/A N/A p is_even N/A N/A N/A p N/A dihedral N/A N/A n N/A N/A N/A 
    Generates the dihedral group of order 2n, Dn.

    The result is given as a subgroup of Sn, except for the special cases n=1
    (the group S2) and n=2 (the Klein 4-group) where that's not possible
    and embeddings in S2 and S4 respectively are given.

    Examples
    ========

    >>> from sympy.combinatorics.permutations import Permutation
    >>> Permutation.print_cyclic = True
    >>> from sympy.combinatorics.generators import dihedral
    >>> list(dihedral(3))
    [(2), (0 2), (0 1 2), (1 2), (0 2 1), (2)(0 1)]

    See Also
    ========
    cyclic
     N/A N/A n 1 N/A N/A N/A N/A Permutation N/A 0 1 N/A N/A N/A Permutation N/A 1 0 N/A N/A N/A n 2 N/A N/A N/A N/A Permutation N/A 0 1 2 3 N/A N/A N/A Permutation N/A 1 0 3 2 N/A N/A N/A Permutation N/A 2 3 0 1 N/A N/A N/A Permutation N/A 3 2 1 0 N/A N/A gen N/A list N/A range n N/A i N/A range n N/A N/A N/A N/A Permutation gen N/A N/A N/A Permutation N/A gen N/A -1 N/A gen N/A rotate_left gen 1 N/A rubik_cube_generators N/A N/A N/A N/A N/A Return the permutations of the 3x3 Rubik's cube, see
    http://www.gap-system.org/Doc/Examples/rubik.html
     N/A a N/A N/A N/A 1 3 8 6 N/A 2 5 7 4 N/A 9 33 25 17 N/A 10 34 26 18 N/A 11 35 27 19 N/A N/A 9 11 16 14 N/A 10 13 15 12 N/A 1 17 41 40 N/A 4 20 44 37 N/A 6 22 46 35 N/A N/A 17 19 24 22 N/A 18 21 23 20 N/A 6 25 43 16 N/A 7 28 42 13 N/A 8 30 41 11 N/A N/A 25 27 32 30 N/A 26 29 31 28 N/A 3 38 43 19 N/A 5 36 45 21 N/A 8 33 48 24 N/A N/A 33 35 40 38 N/A 34 37 39 36 N/A 3 9 46 32 N/A 2 12 47 29 N/A 1 14 48 27 N/A N/A 41 43 48 46 N/A 42 45 47 44 N/A 14 22 30 38 N/A 15 23 31 39 N/A 16 24 32 40 N/A N/A N/A Permutation N/A N/A N/A i 1 N/A i xi N/A xi x size 48 N/A x a N/A rubik N/A N/A n N/A N/A N/A Return permutations for an nxn Rubik's cube.

    Permutations returned are for rotation of each of the slice
    from the face up to the last face for each of the 3 sides (in this order):
    front, right and bottom. Hence, the first n - 1 permutations are for the
    slices from the front.
     N/A N/A n 2 N/A N/A N/A ValueError dimension of cube must be > 1 getr N/A N/A f i N/A N/A N/A N/A N/A N/A faces N/A f col N/A n i N/A getl N/A N/A f i N/A N/A N/A N/A N/A N/A faces N/A f col N/A i 1 N/A getu N/A N/A f i N/A N/A N/A N/A N/A N/A faces N/A f row N/A i 1 N/A getd N/A N/A f i N/A N/A N/A N/A N/A N/A faces N/A f row N/A n i N/A setr N/A N/A f i s N/A N/A N/A N/A N/A faces N/A f N/A N/A N/A N/A n i N/A Matrix n 1 s N/A setl N/A N/A f i s N/A N/A N/A N/A N/A faces N/A f N/A N/A N/A N/A i 1 N/A Matrix n 1 s N/A setu N/A N/A f i s N/A N/A N/A N/A N/A faces N/A f N/A N/A N/A i 1 N/A N/A Matrix 1 n s N/A setd N/A N/A f i s N/A N/A N/A N/A N/A faces N/A f N/A N/A N/A n i N/A N/A Matrix 1 n s N/A cw N/A N/A F r N/A 1 N/A N/A _ N/A range r N/A N/A face N/A faces N/A F N/A rv N/A N/A c N/A range n N/A N/A r N/A range N/A n 1 -1 -1 N/A N/A N/A N/A rv append N/A face N/A N/A r c N/A N/A faces N/A F N/A Matrix n n rv N/A ccw N/A N/A F N/A N/A N/A N/A cw F 3 N/A fcw N/A N/A i r N/A 1 N/A N/A _ N/A range r N/A N/A N/A i 0 N/A N/A N/A cw F N/A i 1 N/A temp N/A getr L i N/A N/A setr L i N/A list N/A getu D i N/A N/A setu D i N/A list N/A reversed N/A getl R i N/A N/A setl R i N/A list N/A getd U i N/A N/A setd U i N/A list N/A reversed temp N/A i 1 N/A fccw N/A N/A i N/A N/A N/A N/A fcw i 3 N/A FCW N/A N/A r N/A 1 N/A N/A _ N/A range r N/A N/A N/A cw F N/A N/A ccw B N/A N/A cw U N/A t N/A faces N/A U N/A N/A cw L N/A N/A faces N/A U N/A faces N/A L N/A N/A cw D N/A N/A faces N/A L N/A faces N/A D N/A N/A cw R N/A N/A faces N/A D N/A faces N/A R N/A N/A faces N/A R t N/A FCCW N/A N/A N/A N/A N/A N/A FCW 3 N/A UCW N/A N/A r N/A 1 N/A N/A _ N/A range r N/A N/A N/A cw U N/A N/A ccw D N/A t N/A faces N/A F N/A N/A faces N/A F N/A faces N/A R N/A N/A faces N/A R N/A faces N/A B N/A N/A faces N/A B N/A faces N/A L N/A N/A faces N/A L t N/A UCCW N/A N/A N/A N/A N/A N/A UCW 3 N/A N/A N/A U F R B L D names N/A symbols U, F, R, B, L, D N/A faces N/A N/A count 0 N/A fi N/A range 6 N/A N/A f N/A N/A a N/A range N/A n 2 N/A N/A N/A N/A f append count N/A count 1 N/A N/A faces N/A N/A names N/A fi N/A Matrix n n f perm N/A N/A show N/A 0 N/A N/A p N/A N/A f names N/A N/A N/A N/A p extend N/A faces N/A f N/A show N/A N/A p N/A N/A N/A g append N/A Permutation p N/A N/A g N/A N/A I N/A list N/A range N/A 6 N/A n 2 N/A i N/A range N/A n 1 N/A N/A N/A fcw i N/A N/A perm N/A N/A fccw i N/A N/A N/A perm 1 I N/A N/A UCW N/A i N/A range N/A n 1 N/A N/A N/A fcw i N/A N/A UCCW N/A N/A perm N/A N/A UCW N/A N/A fccw i N/A N/A UCCW N/A N/A N/A perm 1 I N/A N/A FCW N/A N/A UCCW N/A N/A FCCW N/A i N/A range N/A n 1 N/A N/A N/A fcw i N/A N/A FCW N/A N/A UCW N/A N/A FCCW N/A N/A perm N/A N/A FCW N/A N/A UCCW N/A N/A FCCW N/A N/A fccw i N/A N/A FCW N/A N/A UCW N/A N/A FCCW N/A N/A N/A perm 1 I N/A g N/A