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dyck_word.py
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dyck_word.py
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# -*- coding: utf-8 -*-
r"""
Dyck Words
A class of an object enumerated by the
:func:`Catalan numbers<sage.combinat.combinat.catalan_number>`,
see [Sta-EC2]_, [StaCat98]_ for details.
AUTHORS:
- Mike Hansen
- Dan Drake (2008--05-30): DyckWordBacktracker support
- Florent Hivert (2009--02-01): Bijections with NonDecreasingParkingFunctions
- Christian Stump (2011--12): added combinatorial maps and statistics
- Mike Zabrocki:
* (2012--10): added pretty print, characteristic function, more functions
* (2013--01): added inverse of area/dinv, bounce/area map
- Jean--Baptiste Priez, Travis Scrimshaw (2013--05-17): Added ASCII art
- Travis Scrimshaw (2013--07-09): Removed ``CombinatorialClass`` and added
global options.
REFERENCES:
.. [Sta-EC2] Richard P. Stanley.
*Enumerative Combinatorics*, Volume 2.
Cambridge University Press, 2001.
.. [StaCat98] Richard Stanley. *Exercises on Catalan and Related Numbers
excerpted from Enumerative Combinatorics, vol. 2 (CUP 1999)*,
version of 23 June 1998.
http://www-math.mit.edu/~rstan/ec/catalan.pdf
.. [Hag2008] James Haglund. *The* `q,t` -- *Catalan Numbers and the
Space of Diagonal Harmonics:
With an Appendix on the Combinatorics of Macdonald Polynomials*.
University of Pennsylvania, Philadelphia -- AMS, 2008, 167 pp.
"""
#*****************************************************************************
# Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>,
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# The full text of the GPL is available at:
#
# http://www.gnu.org/licenses/
#*****************************************************************************
from combinat import CombinatorialElement, catalan_number
from sage.combinat.combinatorial_map import combinatorial_map
from backtrack import GenericBacktracker
from sage.structure.global_options import GlobalOptions
from sage.structure.parent import Parent
from sage.structure.unique_representation import UniqueRepresentation
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
from sage.categories.all import Posets
from sage.rings.all import ZZ, QQ
from sage.combinat.permutation import Permutation, Permutations
from sage.combinat.words.word import Word
from sage.combinat.alternating_sign_matrix import AlternatingSignMatrices
from sage.misc.latex import latex
from sage.misc.superseded import deprecated_function_alias
DyckWordOptions = GlobalOptions(name='Dyck words',
doc=r"""
Set and display the global options for Dyck words. If no parameters
are set, then the function returns a copy of the options dictionary.
The ``options`` to Dyck words can be accessed as the method
:obj:`DyckWords.global_options` of :class:`DyckWords` and
related parent classes.
""",
end_doc=r"""
EXAMPLES::
sage: D = DyckWord([1, 1, 0, 1, 0, 0])
sage: D
[1, 1, 0, 1, 0, 0]
sage: DyckWords.global_options(display="lattice")
sage: D
___
_| x
| x .
| . .
sage: DyckWords.global_options(diagram_style="line")
sage: D
/\/\
/ \
sage: DyckWords.global_options.reset()
""",
display=dict(default="list",
description='Specifies how Dyck words should be printed',
values=dict(list='displayed as a list',
lattice='displayed on the lattice defined by ``diagram_style``'),
case_sensitive=False),
ascii_art=dict(default="path",
description='Specifies how the ascii art of Dyck words should be printed',
values=dict(path="Using the path string",
pretty_output="Using pretty printing"),
alias=dict(pretty_print="pretty_output", path_string="path"),
case_sensitive=False),
diagram_style=dict(default="grid",
values=dict(grid='printing as paths on a grid using N and E steps',
line='printing as paths on a line using NE and SE steps',),
alias={'N-E': 'grid', 'NE-SE': 'line'},
case_sensitive=False),
latex_tikz_scale=dict(default=1,
description='The default value for the tikz scale when latexed',
checker=lambda x: True), # More trouble than it's worth to check
latex_diagonal=dict(default=False,
description='The default value for displaying the diagonal when latexed',
checker=lambda x: isinstance(x, bool)),
latex_line_width_scalar=dict(default=2,
description='The default value for the line width as a'
'multiple of the tikz scale when latexed',
checker=lambda x: True), # More trouble than it's worth to check
latex_color=dict(default="black",
description='The default value for the color when latexed',
checker=lambda x: isinstance(x, str)),
latex_bounce_path=dict(default=False,
description='The default value for displaying the bounce path when latexed',
checker=lambda x: isinstance(x, bool)),
latex_peaks=dict(default=False,
description='The default value for displaying the peaks when latexed',
checker=lambda x: isinstance(x, bool)),
latex_valleys=dict(default=False,
description='The default value for displaying the valleys when latexed',
checker=lambda x: isinstance(x, bool)),
)
open_symbol = 1
close_symbol = 0
def replace_parens(x):
r"""
A map sending ``'('`` to ``open_symbol`` and ``')'`` to
``close_symbol``, and raising an error on any input other than
``'('`` and ``')'``. The values of the constants ``open_symbol``
and ``close_symbol`` are subject to change.
This is the inverse map of :func:`replace_symbols`.
INPUT:
- ``x`` -- either an opening or closing parenthesis
OUTPUT:
- If ``x`` is an opening parenthesis, replace ``x`` with the
constant ``open_symbol``.
- If ``x`` is a closing parenthesis, replace ``x`` with the
constant ``close_symbol``.
- Raise a ``ValueError`` if ``x`` is neither an opening nor a
closing parenthesis.
.. SEEALSO:: :func:`replace_symbols`
EXAMPLES::
sage: from sage.combinat.dyck_word import replace_parens
sage: replace_parens('(')
1
sage: replace_parens(')')
0
sage: replace_parens(1)
Traceback (most recent call last):
...
ValueError
"""
if x == '(':
return open_symbol
elif x == ')':
return close_symbol
else:
raise ValueError
def replace_symbols(x):
r"""
A map sending ``open_symbol`` to ``'('`` and ``close_symbol`` to ``')'``,
and raising an error on any input other than ``open_symbol`` and
``close_symbol``. The values of the constants ``open_symbol``
and ``close_symbol`` are subject to change.
This is the inverse map of :func:`replace_parens`.
INPUT:
- ``x`` -- either ``open_symbol`` or ``close_symbol``.
OUTPUT:
- If ``x`` is ``open_symbol``, replace ``x`` with ``'('``.
- If ``x`` is ``close_symbol``, replace ``x`` with ``')'``.
- If ``x`` is neither ``open_symbol`` nor ``close_symbol``, a
``ValueError`` is raised.
.. SEEALSO:: :func:`replace_parens`
EXAMPLES::
sage: from sage.combinat.dyck_word import replace_symbols
sage: replace_symbols(1)
'('
sage: replace_symbols(0)
')'
sage: replace_symbols(3)
Traceback (most recent call last):
...
ValueError
"""
if x == open_symbol:
return '('
elif x == close_symbol:
return ')'
else:
raise ValueError
class DyckWord(CombinatorialElement):
r"""
A Dyck word.
A Dyck word is a sequence of open and close symbols such that every close
symbol has a corresponding open symbol preceding it. That is to say, a
Dyck word of length `n` is a list with `k` entries 1 and `n - k`
entries 0 such that the first `i` entries always have at least as many 1s
among them as 0s. (Here, the 1 serves as the open symbol and the 0 as the
close symbol.) Alternatively, the alphabet 1 and 0 can be replaced by
other characters such as '(' and ')'.
A Dyck word is *complete* if every open symbol moreover has a corresponding
close symbol.
A Dyck word may also be specified by either a noncrossing partition or
by an area sequence or the sequence of heights.
A Dyck word may also be thought of as a lattice path in the `\mathbb{Z}^2`
grid, starting at the origin `(0,0)`, and with steps in the North
`N = (0,1)` and east `E = (1,0)` directions such that it does not pass
below the `x = y` diagonal. The diagonal is referred to as the "main
diagonal" in the documentation. A North step is represented by a 1 in
the list and an East step is represented by a 0.
Equivalently, the path may be represented with steps in
the `NE = (1,1)` and the `SE = (1,-1)` direction such that it does not
pass below the horizontal axis.
A path representing a Dyck word (either using `N` and `E` steps, or
using `NE` and `SE` steps) is called a Dyck path.
EXAMPLES::
sage: dw = DyckWord([1, 0, 1, 0]); dw
[1, 0, 1, 0]
sage: print dw
()()
sage: print dw.height()
1
sage: dw.to_noncrossing_partition()
[[1], [2]]
::
sage: DyckWord('()()')
[1, 0, 1, 0]
sage: DyckWord('(())')
[1, 1, 0, 0]
sage: DyckWord('((')
[1, 1]
sage: DyckWord('')
[]
::
sage: DyckWord(noncrossing_partition=[[1],[2]])
[1, 0, 1, 0]
sage: DyckWord(noncrossing_partition=[[1,2]])
[1, 1, 0, 0]
sage: DyckWord(noncrossing_partition=[])
[]
::
sage: DyckWord(area_sequence=[0,0])
[1, 0, 1, 0]
sage: DyckWord(area_sequence=[0,1])
[1, 1, 0, 0]
sage: DyckWord(area_sequence=[0,1,2,2,0,1,1,2])
[1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0]
sage: DyckWord(area_sequence=[])
[]
::
sage: DyckWord(heights_sequence=(0,1,0,1,0))
[1, 0, 1, 0]
sage: DyckWord(heights_sequence=(0,1,2,1,0))
[1, 1, 0, 0]
sage: DyckWord(heights_sequence=(0,))
[]
::
sage: print DyckWord([1,0,1,1,0,0]).to_path_string()
/\
/\/ \
sage: DyckWord([1,0,1,1,0,0]).pretty_print()
___
| x
_| .
| . .
"""
@staticmethod
def __classcall_private__(cls, dw=None, noncrossing_partition=None,
area_sequence=None, heights_sequence=None,
catalan_code=None):
"""
Return an element with the appropriate parent.
EXAMPLES::
sage: DyckWord([1,0,1,1,0,0])
[1, 0, 1, 1, 0, 0]
sage: DyckWord(heights_sequence=(0,1,2,1,0))
[1, 1, 0, 0]
sage: DyckWord(noncrossing_partition=[[1],[2]])
[1, 0, 1, 0]
"""
if dw is None:
if catalan_code is not None:
return CompleteDyckWords_all().from_Catalan_code(catalan_code)
if area_sequence is not None:
return CompleteDyckWords_all().from_area_sequence(area_sequence)
if noncrossing_partition is not None:
return CompleteDyckWords_all().from_noncrossing_partition(noncrossing_partition)
if heights_sequence is not None:
if heights_sequence[-1] == 0:
P = CompleteDyckWords_all()
else:
P = DyckWords_all()
return P.from_heights(heights_sequence)
raise ValueError("You have not specified a Dyck word.")
if isinstance(dw, str):
l = [replace_parens(_) for _ in dw]
else:
l = dw
if isinstance(l, DyckWord):
return l
# CS: what happens here? there is a loop after a return (which is thus never used)
#elif l in DyckWords() or is_a(l):
#return DyckWord(l)
#for opt in l._latex_options:
#if opt not in latex_options:
#latex_options[opt] = l._latex_options[opt]
#return DyckWord(l,latex_options=latex_options)
if l in CompleteDyckWords_all():
return CompleteDyckWords_all()(l)
if is_a(l):
return DyckWords_all()(l)
raise ValueError("invalid Dyck word")
def __init__(self, parent, l, latex_options={}):
r"""
TESTS::
sage: DW = DyckWords(complete=False).from_heights((0,))
sage: TestSuite(DW).run()
sage: DW = DyckWords(complete=False).min_from_heights((0,))
sage: TestSuite(DW).run()
sage: DW = DyckWords().from_Catalan_code([])
sage: TestSuite(DW).run()
sage: DW = DyckWords().from_area_sequence([])
sage: TestSuite(DW).run()
"""
CombinatorialElement.__init__(self, parent, l)
self._latex_options = dict(latex_options)
_has_2D_print = False
def set_latex_options(self, D):
r"""
Set the latex options for use in the ``_latex_`` function. The
default values are set in the ``__init__`` function.
- ``tikz_scale`` -- (default: 1) scale for use with the tikz package.
- ``diagonal`` -- (default: ``False``) boolean value to draw the
diagonal or not.
- ``line width`` -- (default: 2*``tikz_scale``) value representing the
line width.
- ``color`` -- (default: black) the line color.
- ``bounce path`` -- (default: ``False``) boolean value to indicate
if the bounce path should be drawn.
- ``peaks`` -- (default: ``False``) boolean value to indicate if the
peaks should be displayed.
- ``valleys`` -- (default: ``False``) boolean value to indicate if the
valleys should be displayed.
INPUT:
- ``D`` -- a dictionary with a list of latex parameters to change
EXAMPLES::
sage: D = DyckWord([1,0,1,0,1,0])
sage: D.set_latex_options({"tikz_scale":2})
sage: D.set_latex_options({"valleys":True, "color":"blue"})
"""
for opt in D:
self._latex_options[opt] = D[opt]
def latex_options(self):
r"""
Return the latex options for use in the ``_latex_`` function as a
dictionary. The default values are set using the global options.
- ``tikz_scale`` -- (default: 1) scale for use with the tikz package.
- ``diagonal`` -- (default: ``False``) boolean value to draw the
diagonal or not.
- ``line width`` -- (default: 2*``tikz_scale``) value representing the
line width.
- ``color`` -- (default: black) the line color.
- ``bounce path`` -- (default: ``False``) boolean value to indicate
if the bounce path should be drawn.
- ``peaks`` -- (default: ``False``) boolean value to indicate if the
peaks should be displayed.
- ``valleys`` -- (default: ``False``) boolean value to indicate if the
valleys should be displayed.
EXAMPLES::
sage: D = DyckWord([1,0,1,0,1,0])
sage: D.latex_options()
{'bounce path': False,
'color': 'black',
'diagonal': False,
'line width': 2,
'peaks': False,
'tikz_scale': 1,
'valleys': False}
"""
d = self._latex_options.copy()
if "tikz_scale" not in d:
d["tikz_scale"] = self.parent().global_options["latex_tikz_scale"]
if "diagonal" not in d:
d["diagonal"] = self.parent().global_options["latex_diagonal"]
if "line width" not in d:
d["line width"] = self.parent().global_options["latex_line_width_scalar"]*d["tikz_scale"]
if "color" not in d:
d["color"] = self.parent().global_options["latex_color"]
if "bounce path" not in d:
d["bounce path"] = self.parent().global_options["latex_bounce_path"]
if "peaks" not in d:
d["peaks"] = self.parent().global_options["latex_peaks"]
if "valleys" not in d:
d["valleys"] = self.parent().global_options["latex_valleys"]
return d
def _repr_(self):
r"""
TESTS::
sage: DyckWord([1, 0, 1, 0])
[1, 0, 1, 0]
sage: DyckWord([1, 1, 0, 0])
[1, 1, 0, 0]
sage: type(DyckWord([]))._has_2D_print = True
sage: DyckWord([1, 0, 1, 0])
/\/\
sage: DyckWord([1, 1, 0, 0])
/\
/ \
sage: type(DyckWord([]))._has_2D_print = False
"""
if self._has_2D_print:
return self.to_path_string()
else:
return super(DyckWord, self)._repr_()
def _repr_lattice(self, type=None, labelling=None, underpath=True):
r"""
See :meth:`pretty_print()`.
TESTS::
sage: print DyckWord(area_sequence=[0,1,0])._repr_lattice(type="NE-SE")
/\
/ \/\
sage: print DyckWord(area_sequence=[0,1,0])._repr_lattice(labelling=[1,3,2],underpath=False)
_
___| 2
| x . 3
| . . 1
"""
if type is None:
type = self.parent().global_options['diagram_style']
if type == "grid":
type = "N-E"
elif type == "line":
type = "NE-SE"
if type == "NE-SE":
if labelling is not None or underpath is not True:
raise ValueError("The labelling cannot be shown with Northeast-Southeast paths.")
return self.to_path_string()
elif type == "N-E":
alst = self.to_area_sequence()
n = len(alst)
if n == 0:
return ".\n"
if labelling is None:
labels = [" "]*n
else:
if len(labelling) != n:
raise ValueError("The given labelling has the wrong length.")
labels = [str(label) for label in labelling]
if not underpath:
max_length = max(len(label) for label in labels)
labels = [lbl.rjust(max_length + 1) for lbl in labels]
length_of_final_fall = list(reversed(self)).index(open_symbol)
if length_of_final_fall == 0:
final_fall = " "
else:
final_fall = " _" + "__"*(length_of_final_fall-1)
row = " "*(n - alst[-1]-1) + final_fall + "\n"
for i in range(n - 1):
c = 0
row = row + " "*(n-i-2-alst[-i-2])
c += n-i-2-alst[-i-2]
if alst[-i-2]+1 != alst[-i-1]:
row += " _"
c += alst[-i-2] - alst[-i-1]
if underpath:
row += "__"*(alst[-i-2]-alst[-i-1])+"|" + labels[-1] + "x "*(n-c-2-i) + " ."*i + "\n"
else:
row += "__"*(alst[-i-2]-alst[-i-1])+"| " + "x "*(n-c-2-i) + " ."*i + labels[-1] + "\n"
labels.pop()
if underpath:
row += "|" + labels[-1] + " ."*(n-1) + "\n"
else:
row += "| "+" ."*(n-1) + labels[-1] + "\n"
return row
else:
raise ValueError("The given type (=\s) is not valid." % type)
@staticmethod
def set_ascii_art(rep="path"):
r"""
TESTS::
sage: DyckWord.set_ascii_art("path")
doctest:...: DeprecationWarning: set_ascii_art is deprecated. Use DyckWords.global_options instead.
See http://trac.sagemath.org/14875 for details.
"""
from sage.misc.superseded import deprecation
deprecation(14875, 'set_ascii_art is deprecated. Use DyckWords.global_options instead.')
DyckWords.global_options(ascii_art=rep)
def _ascii_art_(self):
r"""
Return an ASCII art representation of ``self``.
TESTS::
sage: ascii_art(list(DyckWords(3)))
[ /\ ]
[ /\ /\ /\/\ / \ ]
[ /\/\/\, /\/ \, / \/\, / \, / \ ]
"""
from sage.typeset.ascii_art import AsciiArt
rep = self.parent().global_options['ascii_art']
if rep == "path":
ret = self.to_path_string()
elif rep == "pretty_output":
ret = self._repr_lattice()
return AsciiArt(ret.splitlines(), baseline=0)
def _unicode_art_(self):
r"""
Return an unicode art representation of this Dyck word.
EXAMPLES::
sage: unicode_art(list(DyckWords(3)))
⎡ ╱╲ ⎤
⎢ ╱╲ ╱╲ ╱╲╱╲ ╱ ╲ ⎥
⎣ ╱╲╱╲╱╲, ╱╲╱ ╲, ╱ ╲╱╲, ╱ ╲, ╱ ╲ ⎦
"""
from sage.typeset.unicode_art import UnicodeArt
return UnicodeArt(self.to_path_string(unicode=True).splitlines())
def __str__(self):
r"""
Return a string consisting of matched parentheses corresponding to
the Dyck word.
EXAMPLES::
sage: print DyckWord([1, 0, 1, 0])
()()
sage: print DyckWord([1, 1, 0, 0])
(())
"""
if self._has_2D_print:
return self.to_path_string()
else:
return "".join(map(replace_symbols, [x for x in self]))
def to_path_string(self, unicode=False):
r"""
A path representation of the Dyck word consisting of steps
``/`` and ``\`` .
EXAMPLES::
sage: print DyckWord([1, 0, 1, 0]).to_path_string()
/\/\
sage: print DyckWord([1, 1, 0, 0]).to_path_string()
/\
/ \
sage: print DyckWord([1,1,0,1,1,0,0,1,0,1,0,0]).to_path_string()
/\
/\/ \/\/\
/ \
"""
if unicode:
import unicodedata
space = u' '
up = unicodedata.lookup('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT')
down = unicodedata.lookup('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT')
else:
space = ' '
up = '/'
down = '\\'
res = [([space]*len(self)) for _ in range(self.height())]
h = 1
for i, p in enumerate(self):
if p == open_symbol:
res[-h][i] = up
h += 1
else:
h -= 1
res[-h][i] = down
return "\n".join("".join(l) for l in res)
def pretty_print(self, type=None, labelling=None, underpath=True):
r"""
Display a DyckWord as a lattice path in the `\ZZ^2` grid.
If the ``type`` is "N-E", then the a cell below the diagonal is
indicated by a period, whereas a cell below the path but above
the diagonal is indicated by an x. If a list of labels is
included, they are displayed along the vertical edges of the
Dyck path.
If the ``type`` is "NE-SE", then the path is simply printed
as up steps and down steps.
INPUT:
- ``type`` -- (default: ``None``) can either be:
- ``None`` to use the global option default
- "N-E" to show ``self`` as a path of north and east steps, or
- "NE-SE" to show ``self`` as a path of north-east and
south-east steps.
- ``labelling`` -- (if type is "N-E") a list of labels assigned to
the up steps in ``self``.
- ``underpath`` -- (if type is "N-E", default:``True``) If ``True``,
the labelling is shown under the path; otherwise, it is shown to
the right of the path.
EXAMPLES::
sage: for D in DyckWords(3): D.pretty_print()
_
_|
_| .
| . .
___
| x
_| .
| . .
_
___|
| x .
| . .
___
_| x
| x .
| . .
_____
| x x
| x .
| . .
::
sage: for D in DyckWords(3): D.pretty_print(type="NE-SE")
/\/\/\
/\
/\/ \
/\
/ \/\
/\/\
/ \
/\
/ \
/ \
::
sage: D = DyckWord([1,1,1,0,1,0,0,1,1])
sage: D.pretty_print()
| x x
___| x .
_| x x . .
| x x . . .
| x . . . .
| . . . . .
sage: D = DyckWord([1,1,1,0,1,0,0,1,1,0])
sage: D.pretty_print()
_
| x x
___| x .
_| x x . .
| x x . . .
| x . . . .
| . . . . .
sage: D = DyckWord([1,1,1,0,1,0,0,1,1,0,0])
sage: D.pretty_print()
___
| x x
___| x .
_| x x . .
| x x . . .
| x . . . .
| . . . . .
::
sage: DyckWord(area_sequence=[0,1,0]).pretty_print(labelling=[1,3,2])
_
___|2
|3x .
|1 . .
sage: DyckWord(area_sequence=[0,1,0]).pretty_print(labelling=[1,3,2],underpath=False)
_
___| 2
| x . 3
| . . 1
::
sage: DyckWord(area_sequence=[0,1,1,2,3,2,3,3,2,0,1,1,2,3,4,2,3]).pretty_print()
_______
| x x x
_____| x x .
| x x x x . .
| x x x . . .
| x x . . . .
_| x . . . . .
| x . . . . . .
_____| . . . . . . .
___| x x . . . . . . . .
_| x x x . . . . . . . . .
| x x x . . . . . . . . . .
___| x x . . . . . . . . . . .
| x x x . . . . . . . . . . . .
| x x . . . . . . . . . . . . .
_| x . . . . . . . . . . . . . .
| x . . . . . . . . . . . . . . .
| . . . . . . . . . . . . . . . .
sage: DyckWord(area_sequence=[0,1,1,2,3,2,3,3,2,0,1,1,2,3,4,2,3]).pretty_print(labelling=range(17),underpath=False)
_______
| x x x 16
_____| x x . 15
| x x x x . . 14
| x x x . . . 13
| x x . . . . 12
_| x . . . . . 11
| x . . . . . . 10
_____| . . . . . . . 9
___| x x . . . . . . . . 8
_| x x x . . . . . . . . . 7
| x x x . . . . . . . . . . 6
___| x x . . . . . . . . . . . 5
| x x x . . . . . . . . . . . . 4
| x x . . . . . . . . . . . . . 3
_| x . . . . . . . . . . . . . . 2
| x . . . . . . . . . . . . . . . 1
| . . . . . . . . . . . . . . . . 0
::
sage: DyckWord([]).pretty_print()
.
"""
print self._repr_lattice(type, labelling, underpath)
pp = pretty_print
def _latex_(self):
r"""
A latex representation of ``self`` using the tikzpicture package.
EXAMPLES::
sage: D = DyckWord([1,0,1,1,1,0,1,1,0,0,0,1,0,0])
sage: D.set_latex_options({"valleys":True, "peaks":True, "bounce path":True})
sage: latex(D)
\vcenter{\hbox{$\begin{tikzpicture}[scale=1]
\draw[line width=2,color=red,fill=red] (2, 0) circle (0.21);
\draw[line width=2,color=red,fill=red] (6, 2) circle (0.21);
\draw[line width=2,color=red,fill=red] (11, 1) circle (0.21);
\draw[line width=2,color=red,fill=red] (1, 1) circle (0.21);
\draw[line width=2,color=red,fill=red] (5, 3) circle (0.21);
\draw[line width=2,color=red,fill=red] (8, 4) circle (0.21);
\draw[line width=2,color=red,fill=red] (12, 2) circle (0.21);
\draw[rounded corners=1, color=green, line width=4] (0, 0) -- (1, 1) -- (2, 0) -- (3, 1) -- (4, 0) -- (5, 1) -- (6, 2) -- (7, 3) -- (8, 2) -- (9, 1) -- (10, 0) -- (11, 1) -- (12, 2) -- (13, 1) -- (14, 0);
\draw[dotted] (0, 0) grid (14, 4);
\draw[rounded corners=1, color=black, line width=2] (0, 0) -- (1, 1) -- (2, 0) -- (3, 1) -- (4, 2) -- (5, 3) -- (6, 2) -- (7, 3) -- (8, 4) -- (9, 3) -- (10, 2) -- (11, 1) -- (12, 2) -- (13, 1) -- (14, 0);
\end{tikzpicture}$}}
sage: DyckWord([1,0])._latex_()
'\\vcenter{\\hbox{$\\begin{tikzpicture}[scale=1]\n \\draw[dotted] (0, 0) grid (2, 1);\n \\draw[rounded corners=1, color=black, line width=2] (0, 0) -- (1, 1) -- (2, 0);\n\\end{tikzpicture}$}}'
sage: DyckWord([1,0,1,1,0,0])._latex_()
'\\vcenter{\\hbox{$\\begin{tikzpicture}[scale=1]\n \\draw[dotted] (0, 0) grid (6, 2);\n \\draw[rounded corners=1, color=black, line width=2] (0, 0) -- (1, 1) -- (2, 0) -- (3, 1) -- (4, 2) -- (5, 1) -- (6, 0);\n\\end{tikzpicture}$}}'
"""
latex.add_package_to_preamble_if_available("tikz")
heights = self.heights()
latex_options = self.latex_options()
diagonal = latex_options["diagonal"]
ht = [(0, 0)]
valleys = []
peaks = []
for i in range(1, len(heights)):
a, b = ht[-1]
if heights[i] > heights[i-1]:
if diagonal:
ht.append((a, b+1))
else:
ht.append((a+1, b+1))
if i < len(heights)-1 and heights[i+1] < heights[i]:
peaks.append(ht[-1])
else:
if diagonal:
ht.append((a+1, b))
else:
ht.append((a+1, b-1))
if i < len(heights)-1 and heights[i+1] > heights[i]:
valleys.append(ht[-1])
ht = iter(ht)
if diagonal:
grid = [((0, i), (i, i+1))
for i in range(self.number_of_open_symbols())]
else:
grid = [((0, 0), (len(self), self.height()))]
res = "\\vcenter{\\hbox{$\\begin{tikzpicture}[scale="+str(latex_options['tikz_scale'])+"]\n"
mark_points = []
if latex_options['valleys']:
mark_points.extend(valleys)
if latex_options['peaks']:
mark_points.extend(peaks)
for v in mark_points:
res += " \\draw[line width=2,color=red,fill=red] %s circle (%s);\n" % (str(v), 0.15 + .03 * latex_options['line width'])
if latex_options["bounce path"]:
D = self.bounce_path()
D.set_latex_options(latex_options)
D.set_latex_options({"color": "green",
"line width": 2 * latex_options['line width'],
"bounce path": False,
"peaks": False, "valleys": False})
res += D._latex_().split("\n")[-2] + "\n"
for v1, v2 in grid:
res += " \\draw[dotted] %s grid %s;\n" % (str(v1), str(v2))
if diagonal:
res += " \\draw (0,0) -- %s;\n" % str((self.number_of_open_symbols(), self.number_of_open_symbols()))
res += " \\draw[rounded corners=1, color=%s, line width=%s] (0, 0)" % (latex_options['color'], str(latex_options['line width']))
next(ht)
for i, j in ht:
res += " -- (%s, %s)" % (i, j)
res += ";\n"
res += "\\end{tikzpicture}$}}"
return res
def length(self):
r"""
Return the length of ``self``.
EXAMPLES::
sage: DyckWord([1, 0, 1, 0]).length()
4
sage: DyckWord([1, 0, 1, 1, 0]).length()
5
TESTS::
sage: DyckWord([]).length()
0
"""
return len(self)
def number_of_open_symbols(self):
r"""
Return the number of open symbols in ``self``.
EXAMPLES::
sage: DyckWord([1, 0, 1, 0]).number_of_open_symbols()
2
sage: DyckWord([1, 0, 1, 1, 0]).number_of_open_symbols()
3
TESTS::
sage: DyckWord([]).number_of_open_symbols()
0
"""
return len([x for x in self if x == open_symbol])
def number_of_close_symbols(self):
r"""
Return the number of close symbols in ``self``.
EXAMPLES::
sage: DyckWord([1, 0, 1, 0]).number_of_close_symbols()
2
sage: DyckWord([1, 0, 1, 1, 0]).number_of_close_symbols()
2
TESTS::
sage: DyckWord([]).number_of_close_symbols()
0
"""
return len([x for x in self if x == close_symbol])
def is_complete(self):
r"""
Return ``True`` if ``self`` is complete.
A Dyck word `d` is complete if `d` contains as many closers as openers.
EXAMPLES::
sage: DyckWord([1, 0, 1, 0]).is_complete()
True
sage: DyckWord([1, 0, 1, 1, 0]).is_complete()
False
TESTS::