def __init__(self, shape, pi=None):
"""
EXAMPLES::
sage: from sage.combinat.sf.ns_macdonald import NonattackingBacktracker
sage: n = NonattackingBacktracker(LatticeDiagram([0,1,2]))
sage: n._ending_position
(3, 2)
sage: n._initial_state
(2, 1)
"""
self._shape = shape
self._n = sum(shape)
self._initial_data = [ [None]*s for s in shape ]
if pi is None:
pi=Permutation([1]).to_permutation_group_element()
self.pi=pi
#The ending position will be at the highest box
#which is farthest right
ending_row = max(shape)
ending_col = len(shape) - list(reversed(list(shape))).index(ending_row)
self._ending_position = (ending_col, ending_row)
#Get the lowest box that is farthest left
starting_row = 1
nonzero = [i for i in shape if i != 0]
starting_col = list(shape).index(nonzero[0]) + 1
GenericBacktracker.__init__(self, self._initial_data, (starting_col, starting_row))
def __init__(self, shape, pi=None):
"""
EXAMPLES::
sage: from sage.combinat.sf.ns_macdonald import NonattackingBacktracker
sage: n = NonattackingBacktracker(LatticeDiagram([0,1,2]))
sage: n._ending_position
(3, 2)
sage: n._initial_state
(2, 1)
"""
self._shape = shape
self._n = sum(shape)
self._initial_data = [[None] * s for s in shape]
if pi == None:
pi = Permutation([1]).to_permutation_group_element()
self.pi = pi
#The ending position will be at the highest box
#which is farthest right
ending_row = max(shape)
ending_col = len(shape) - list(reversed(list(shape))).index(ending_row)
self._ending_position = (ending_col, ending_row)
#Get the lowest box that is farthest left
starting_row = 1
nonzero = [i for i in shape if i != 0]
starting_col = list(shape).index(nonzero[0]) + 1
GenericBacktracker.__init__(self, self._initial_data,
(starting_col, starting_row))
def __init__(self, shape, max_entry=None):
"""
EXAMPLES::
sage: from sage.combinat.composition_tableau import CompositionTableauxBacktracker
sage: n = CompositionTableauxBacktracker([1,3,2])
sage: n._ending_position
(2, 1)
sage: n._initial_state
(0, 0)
"""
self._shape = shape
self._n = sum(shape)
self._initial_data = [[None] * s for s in shape]
if max_entry is None:
max_entry = sum(shape)
self.max_entry = max_entry
# The ending position will be at the lowest box which is farthest right
ending_row = len(shape) - 1
ending_col = shape[-1] - 1
self._ending_position = (ending_row, ending_col)
# Get the highest box that is farthest left
starting_row = 0
starting_col = 0
GenericBacktracker.__init__(self, self._initial_data,
(starting_row, starting_col))
def __init__(self, shape, max_entry=None):
"""
EXAMPLES::
sage: from sage.combinat.composition_tableau import CompositionTableauxBacktracker
sage: n = CompositionTableauxBacktracker([1,3,2])
sage: n._ending_position
(2, 1)
sage: n._initial_state
(0, 0)
"""
self._shape = shape
self._n = sum(shape)
self._initial_data = [ [None]*s for s in shape ]
if max_entry is None:
max_entry=sum(shape)
self.max_entry=max_entry
# The ending position will be at the lowest box which is farthest right
ending_row = len(shape)-1
ending_col = shape[-1]-1
self._ending_position = (ending_row, ending_col)
# Get the highest box that is farthest left
starting_row = 0
starting_col = 0
GenericBacktracker.__init__(self, self._initial_data, (starting_row, starting_col))
def __init__(self, crystal, index_set=None):
r"""
Time complexity: `O(nF)` amortized for each produced
element, where `n` is the size of the index set, and `F` is
the cost of computing `e` and `f` operators.
Memory complexity: `O(D)` where `D` is the depth of the crystal.
Principle of the algorithm:
Let `C` be a classical crystal. It's an acyclic graph where each
connected component has a unique element without predecessors (the
highest weight element for this component). Let's assume for
simplicity that `C` is irreducible (i.e. connected) with highest
weight element `u`.
One can define a natural spanning tree of `C` by taking
`u` as the root of the tree, and for any other element
`y` taking as ancestor the element `x` such that
there is an `i`-arrow from `x` to `y` with
`i` minimal. Then, a path from `u` to `y`
describes the lexicographically smallest sequence
`i_1,\dots,i_k` such that
`(f_{i_k} \circ f_{i_1})(u)=y`.
Morally, the iterator implemented below just does a depth first
search walk through this spanning tree. In practice, this can be
achieved recursively as follows: take an element `x`, and
consider in turn each successor `y = f_i(x)`, ignoring
those such that `y = f_j(x^{\prime})` for some `x^{\prime}` and
`j<i` (this can be tested by computing `e_j(y)`
for `j<i`).
EXAMPLES::
sage: from sage.combinat.crystals.crystals import CrystalBacktracker
sage: C = crystals.Tableaux(['B',3],shape=[3,2,1])
sage: CB = CrystalBacktracker(C)
sage: len(list(CB))
1617
sage: CB = CrystalBacktracker(C, [1,2])
sage: len(list(CB))
8
"""
GenericBacktracker.__init__(self, None, None)
self._crystal = crystal
if index_set is None:
self._index_set = crystal.index_set()
else:
self._index_set = index_set
def __init__(self, crystal, index_set=None):
r"""
Time complexity: `O(nF)` amortized for each produced
element, where `n` is the size of the index set, and `F` is
the cost of computing `e` and `f` operators.
Memory complexity: `O(D)` where `D` is the depth of the crystal.
Principle of the algorithm:
Let `C` be a classical crystal. It's an acyclic graph where all
connected component has a unique element without predecessors (the
highest weight element for this component). Let's assume for
simplicity that `C` is irreducible (i.e. connected) with highest
weight element `u`.
One can define a natural spanning tree of `C` by taking
`u` as the root of the tree, and for any other element
`y` taking as ancestor the element `x` such that
there is an `i`-arrow from `x` to `y` with
`i` minimal. Then, a path from `u` to `y`
describes the lexicographically smallest sequence
`i_1,\dots,i_k` such that
`(f_{i_k} \circ f_{i_1})(u)=y`.
Morally, the iterator implemented below just does a depth first
search walk through this spanning tree. In practice, this can be
achieved recursively as follow: take an element `x`, and
consider in turn each successor `y = f_i(x)`, ignoring
those such that `y = f_j(x^{\prime})` for some `x^{\prime}` and
`j<i` (this can be tested by computing `e_j(y)`
for `j<i`).
EXAMPLES::
sage: from sage.combinat.crystals.crystals import CrystalBacktracker
sage: C = crystals.Tableaux(['B',3],shape=[3,2,1])
sage: CB = CrystalBacktracker(C)
sage: len(list(CB))
1617
sage: CB = CrystalBacktracker(C, [1,2])
sage: len(list(CB))
8
"""
GenericBacktracker.__init__(self, None, None)
self._crystal = crystal
if index_set is None:
self._index_set = crystal.index_set()
else:
self._index_set = index_set