Dyck Words¶

A class of an object enumerated by the Catalan numbers, 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(1,2,3): Richard P. Stanley. Enumerative Combinatorics, Volume 2. Cambridge University Press, 2001.
StaCat98(1,2,3): 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(1,2,3,4,5): 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.
BK2001: J. Bandlow, K. Killpatrick – An area-to_inv bijection between Dyck paths and 312-avoiding permutations, Electronic Journal of Combinatorics, Volume 8, Issue 1 (2001).
EP2004: S. Elizalde, I. Pak. Bijections for refined restricted permutations*. JCTA 105(2) 2004.
CK2008: A. Claesson, S. Kitaev. Classification of bijections between `321`- and `132`- avoiding permutations. Séminaire Lotharingien de Combinatoire 60 2008. arXiv 0805.1325.
Knu1973: D. Knuth. The Art of Computer Programming, Vol. III. Addison-Wesley. Reading, MA. 1973.
Kra2001: C. Krattenthaler – Permutations with restricted patterns and Dyck paths, Adv. Appl. Math. 27 (2001), 510–530.
DS1992: A. Denise, R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math 137 (1992), 155–176.

class sage.combinat.dyck_word.CompleteDyckWords¶

Bases: sage.combinat.dyck_word.DyckWords

Abstract base class for all complete Dyck words.

Element¶: alias of DyckWord_complete

from_Catalan_code(code)¶

Return the Dyck word associated to the given Catalan code code.

A Catalan code of length \(n\) is a sequence \((a_1, a_2, \ldots, a_n)\) of \(n\) integers \(a_i\) such that:

\(0 \leq a_i \leq n-i\) for every \(i\);
if \(i < j\) and \(a_i > 0\) and \(a_j > 0\) and \(a_{i+1} = a_{i+2} = \cdots = a_{j-1} = 0\), then \(a_i - a_j < j-i\).

It turns out that the Catalan codes of length \(n\) are in bijection with Dyck words.

The Catalan code of a Dyck word is example (x) in Richard Stanley’s exercises on combinatorial interpretations for Catalan objects. The code in this example is the reverse of the description provided there. See [Sta-EC2] and [StaCat98].

EXAMPLES:

sage: DyckWords().from_Catalan_code([])
[]
sage: DyckWords().from_Catalan_code([0])
[1, 0]
sage: DyckWords().from_Catalan_code([0, 1])
[1, 1, 0, 0]
sage: DyckWords().from_Catalan_code([0, 0])
[1, 0, 1, 0]

from_area_sequence(code)¶

Return the Dyck word associated to the given area sequence code.

See to_area_sequence() for a definition of the area sequence of a Dyck word.

See also

area(), to_area_sequence().

INPUT:

code – a list of integers satisfying code[0] == 0 and 0 <= code[i+1] <= code[i]+1.

EXAMPLES:

sage: DyckWords().from_area_sequence([])
[]
sage: DyckWords().from_area_sequence([0])
[1, 0]
sage: DyckWords().from_area_sequence([0, 1])
[1, 1, 0, 0]
sage: DyckWords().from_area_sequence([0, 0])
[1, 0, 1, 0]

from_non_decreasing_parking_function(pf)¶

Bijection from non-decreasing parking functions.

See there the method to_dyck_word() for more information.

EXAMPLES:

sage: D = DyckWords()
sage: D.from_non_decreasing_parking_function([])
[]
sage: D.from_non_decreasing_parking_function([1])
[1, 0]
sage: D.from_non_decreasing_parking_function([1,1])
[1, 1, 0, 0]
sage: D.from_non_decreasing_parking_function([1,2])
[1, 0, 1, 0]
sage: D.from_non_decreasing_parking_function([1,1,1])
[1, 1, 1, 0, 0, 0]
sage: D.from_non_decreasing_parking_function([1,2,3])
[1, 0, 1, 0, 1, 0]
sage: D.from_non_decreasing_parking_function([1,1,3,3,4,6,6])
[1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0]

from_noncrossing_partition(ncp)¶: Convert a noncrossing partition ncp to a Dyck word.

class sage.combinat.dyck_word.CompleteDyckWords_all¶

Bases: sage.combinat.dyck_word.CompleteDyckWords, sage.combinat.dyck_word.DyckWords_all

All complete Dyck words.

class height_poset¶

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

The poset of complete Dyck words compared componentwise by heights. This is, D is smaller than or equal to D' if it is weakly below D'.

le(dw1, dw2)¶

Compare two Dyck words of equal size, and return True if all of the heights of dw1 are less than or equal to the respective heights of dw2 .

See also

heights

EXAMPLES:

sage: poset = DyckWords().height_poset()
sage: poset.le(DyckWord([]), DyckWord([]))
True
sage: poset.le(DyckWord([1,0]), DyckWord([1,0]))
True
sage: poset.le(DyckWord([1,0,1,0]), DyckWord([1,1,0,0]))
True
sage: poset.le(DyckWord([1,1,0,0]), DyckWord([1,0,1,0]))
False
sage: [poset.le(dw1, dw2)
....:     for dw1 in DyckWords(3) for dw2 in DyckWords(3)]
[True, True, True, True, True, False, True, False, True, True,
False, False, True, True, True, False, False, False, True,
True, False, False, False, False, True]

class sage.combinat.dyck_word.CompleteDyckWords_size(k)¶

Bases: sage.combinat.dyck_word.CompleteDyckWords, sage.combinat.dyck_word.DyckWords_size

All complete Dyck words of a given size.

cardinality()¶

Return the number of complete Dyck words of semilength \(n\), i.e. the \(n\)-th Catalan number.

EXAMPLES:

sage: DyckWords(4).cardinality()
14
sage: ns = list(range(9))
sage: dws = [DyckWords(n) for n in ns]
sage: all(dw.cardinality() == len(dw.list()) for dw in dws)
True

random_element()¶

Return a random complete Dyck word of semilength \(n\).

The algorithm is based on a classical combinatorial fact. One chooses at random a word with \(n\) 0’s and \(n+1\) 1’s. One then considers every 1 as an ascending step and every 0 as a descending step, and one finds the lowest point of the path (with respect to a slightly tilted slope). One then cuts the path at this point and builds a Dyck word by exchanging the two parts of the word and removing the initial step.

Todo

extend this to m-Dyck words

EXAMPLES:

sage: dw = DyckWords(8)
sage: dw.random_element()  # random
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0]

sage: D = DyckWords(8)
sage: D.random_element() in D
True

class sage.combinat.dyck_word.DyckWord(parent, l, latex_options={})¶

Bases: sage.combinat.combinat.CombinatorialElement

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 \(\ZZ^2\) 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: 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
 _|  .
|  . .

ascent_prime_decomposition()¶

Decompose this Dyck word into a sequence of ascents and prime Dyck paths.

A Dyck word is prime if it is complete and has precisely one return - the final step. In particular, the empty Dyck path is not prime. Thus, the factorization is unique.

This decomposition yields a sequence of odd length: the words with even indices consist of up steps only, the words with odd indices are prime Dyck paths. The concatenation of the result is the original word.

EXAMPLES:

sage: D = DyckWord([1,1,1,0,1,0,1,1,1,1,0,1])
sage: D.ascent_prime_decomposition()
[[1, 1], [1, 0], [], [1, 0], [1, 1, 1], [1, 0], [1]]

sage: DyckWord([]).ascent_prime_decomposition()
[[]]

sage: DyckWord([1,1]).ascent_prime_decomposition()
[[1, 1]]

sage: DyckWord([1,0,1,0]).ascent_prime_decomposition()
[[], [1, 0], [], [1, 0], []]

associated_parenthesis(pos)¶

Report the position for the parenthesis in self that matches the one at position pos.

The positions in self are counted from \(0\).

INPUT:

pos – the index of the parenthesis in the list

OUTPUT:

Integer representing the index of the matching parenthesis. If no parenthesis matches, return None.

EXAMPLES:

sage: DyckWord([1, 0]).associated_parenthesis(0)
1
sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(0)
1
sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(1)
0
sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(2)
3
sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(3)
2
sage: DyckWord([1, 1, 0, 0]).associated_parenthesis(0)
3
sage: DyckWord([1, 1, 0, 0]).associated_parenthesis(2)
1
sage: DyckWord([1, 1, 0]).associated_parenthesis(1)
2
sage: DyckWord([1, 1]).associated_parenthesis(0)

catalan_factorization()¶

Decompose this Dyck word into a sequence of complete Dyck words.

Each element of the list returned is a (possibly empty) complete Dyck word. The original word is obtained by placing an up step between each of these complete Dyck words. Thus, the number of words returned is one more than the final height.

See Section 1.2 of [CC1982] or Lemma 9.1.1 of [Lot2005].

EXAMPLES:

sage: D = DyckWord([1,1,1,0,1,0,1,1,1,1,0,1])
sage: D.catalan_factorization()
[[], [], [1, 0, 1, 0], [], [], [1, 0], []]

sage: DyckWord([]).catalan_factorization()
[[]]

sage: DyckWord([1,1]).catalan_factorization()
[[], [], []]

sage: DyckWord([1,0,1,0]).catalan_factorization()
[[1, 0, 1, 0]]

height()¶

Return the height of self.

We view the Dyck word as a Dyck path from \((0, 0)\) to \((2n, 0)\) in the first quadrant by letting 1’s represent steps in the direction \((1, 1)\) and 0’s represent steps in the direction \((1, -1)\).

The height is the maximum \(y\)-coordinate reached.

See also

heights()

EXAMPLES:

sage: DyckWord([]).height()
0
sage: DyckWord([1,0]).height()
1
sage: DyckWord([1, 1, 0, 0]).height()
2
sage: DyckWord([1, 1, 0, 1, 0]).height()
2
sage: DyckWord([1, 1, 0, 0, 1, 0]).height()
2
sage: DyckWord([1, 0, 1, 0]).height()
1
sage: DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]).height()
3

heights()¶

Return the heights of self.

We view the Dyck word as a Dyck path from \((0,0)\) to \((2n,0)\) in the first quadrant by letting 1’s represent steps in the direction \((1,1)\) and 0’s represent steps in the direction \((1,-1)\).

The heights is the sequence of the \(y\)-coordinates of all \(2n+1\) lattice points along the path.

See also

from_heights(), min_from_heights()

EXAMPLES:

sage: DyckWord([]).heights()
(0,)
sage: DyckWord([1,0]).heights()
(0, 1, 0)
sage: DyckWord([1, 1, 0, 0]).heights()
(0, 1, 2, 1, 0)
sage: DyckWord([1, 1, 0, 1, 0]).heights()
(0, 1, 2, 1, 2, 1)
sage: DyckWord([1, 1, 0, 0, 1, 0]).heights()
(0, 1, 2, 1, 0, 1, 0)
sage: DyckWord([1, 0, 1, 0]).heights()
(0, 1, 0, 1, 0)
sage: DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]).heights()
(0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0)

is_complete()¶

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

latex_options()¶

Return the latex options for use in the _latex_ function as a dictionary.

The default values are set using the 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}

Todo

This should probably be merged into DyckWord.options.

length()¶

Return the length of self.

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).length()
4
sage: DyckWord([1, 0, 1, 1, 0]).length()
5

number_of_close_symbols()¶

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

number_of_double_rises()¶

Return a the number of positions in self where there are two consecutive \(1\)’s.

EXAMPLES:

sage: DyckWord([1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]).number_of_double_rises()
2
sage: DyckWord([1, 1, 0, 0]).number_of_double_rises()
1
sage: DyckWord([1, 0, 1, 0]).number_of_double_rises()
0

number_of_initial_rises()¶

Return the length of the initial run of self.

OUTPUT:

a non–negative integer indicating the length of the initial rise

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).number_of_initial_rises()
1
sage: DyckWord([1, 1, 0, 0]).number_of_initial_rises()
2
sage: DyckWord([1, 1, 0, 0, 1, 0]).number_of_initial_rises()
2
sage: DyckWord([1, 0, 1, 1, 0, 0]).number_of_initial_rises()
1

number_of_open_symbols()¶

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

number_of_peaks()¶

Return the number of peaks of the Dyck path associated to self .

See also

peaks(), number_of_valleys()

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).number_of_peaks()
2
sage: DyckWord([1, 1, 0, 0]).number_of_peaks()
1
sage: DyckWord([1,1,0,1,0,1,0,0]).number_of_peaks()
3
sage: DyckWord([]).number_of_peaks()
0

number_of_touch_points()¶

Return the number of touches of self at the main diagonal.

OUTPUT:

a non–negative integer

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).number_of_touch_points()
2
sage: DyckWord([1, 1, 0, 0]).number_of_touch_points()
1
sage: DyckWord([1, 1, 0, 0, 1, 0]).number_of_touch_points()
2
sage: DyckWord([1, 0, 1, 1, 0, 0]).number_of_touch_points()
2

number_of_valleys()¶

Return the number of valleys of self.

See also

number_of_peaks(), valleys()

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).number_of_valleys()
1
sage: DyckWord([1, 1, 0, 0]).number_of_valleys()
0
sage: DyckWord([1, 1, 0, 0, 1, 0]).number_of_valleys()
1
sage: DyckWord([1, 0, 1, 1, 0, 0]).number_of_valleys()
1

peaks()¶

Return a list of the positions of the peaks of a Dyck word.

A peak is \(1\) followed by a \(0\). Note that this does not agree with the definition given in [Hag2008].

See also

valleys(), number_of_peaks()

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).peaks()
[0, 2]
sage: DyckWord([1, 1, 0, 0]).peaks()
[1]
sage: DyckWord([1,1,0,1,0,1,0,0]).peaks() # Haglund's def gives 2
[1, 3, 5]

plot(**kwds)¶

Plot a Dyck word as a continuous path.

EXAMPLES:

sage: w = DyckWords(100).random_element()
sage: w.plot()
Graphics object consisting of 1 graphics primitive

position_of_first_return()¶

Return the number of vertical steps before the Dyck path returns to the main diagonal.

EXAMPLES:

sage: DyckWord([1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]).position_of_first_return()
1
sage: DyckWord([1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0]).position_of_first_return()
7
sage: DyckWord([1, 1, 0, 0]).position_of_first_return()
2
sage: DyckWord([1, 0, 1, 0]).position_of_first_return()
1
sage: DyckWord([]).position_of_first_return()
0

positions_of_double_rises()¶

Return a list of positions in self where there are two consecutive \(1\)’s.

EXAMPLES:

sage: DyckWord([1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]).positions_of_double_rises()
[2, 5]
sage: DyckWord([1, 1, 0, 0]).positions_of_double_rises()
[0]
sage: DyckWord([1, 0, 1, 0]).positions_of_double_rises()
[]

pp(type=None, labelling=None, underpath=True)¶

Display a DyckWord as a lattice path in the \(\ZZ^2\) grid.

If the type is “N-E”, then 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 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=list(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()
.

pretty_print(type=None, labelling=None, underpath=True)¶

Display a DyckWord as a lattice path in the \(\ZZ^2\) grid.

If the type is “N-E”, then 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 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=list(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()
.

returns_to_zero()¶

Return a list of positions where self has height \(0\), excluding the position \(0\).

EXAMPLES:

sage: DyckWord([]).returns_to_zero()
[]
sage: DyckWord([1, 0]).returns_to_zero()
[2]
sage: DyckWord([1, 0, 1, 0]).returns_to_zero()
[2, 4]
sage: DyckWord([1, 1, 0, 0]).returns_to_zero()
[4]

rise_composition()¶

The sequences of lengths of runs of \(1\)’s in self. Also equal to the sequence of lengths of vertical segments in the Dyck path.

EXAMPLES:

sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).pretty_print()
           ___
          | x
   _______|  .
  | x x x  . .
  | x x  . . .
 _| x  . . . .
| x  . . . . .
|  . . . . . .

sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).rise_composition()
[2, 3, 2]
sage: DyckWord([1,1,0,0]).rise_composition()
[2]
sage: DyckWord([1,0,1,0]).rise_composition()
[1, 1]

set_latex_options(D)¶

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"})

Todo

This should probably be merged into DyckWord.options.

tamari_interval(other)¶

Return the Tamari interval between self and other as a TamariIntervalPoset.

A “Tamari interval” means an interval in the Tamari order. The Tamari order on the set of Dyck words of size \(n\) is the partial order obtained from the Tamari order on the set of binary trees of size \(n\) (see tamari_lequal()) by means of the Tamari bijection between Dyck words and binary trees (to_dyck_word_tamari()).

INPUT:

other – a Dyck word greater or equal to self in the Tamari order

EXAMPLES:

sage: dw = DyckWord([1, 1, 0, 1, 0, 0, 1, 0])
sage: ip = dw.tamari_interval(DyckWord([1, 1, 1, 0, 0, 1, 0, 0])); ip
The Tamari interval of size 4 induced by relations [(2, 4), (3, 4), (3, 1), (2, 1)]
sage: ip.lower_dyck_word()
[1, 1, 0, 1, 0, 0, 1, 0]
sage: ip.upper_dyck_word()
[1, 1, 1, 0, 0, 1, 0, 0]
sage: ip.interval_cardinality()
4
sage: ip.number_of_tamari_inversions()
2
sage: list(ip.dyck_words())
[[1, 1, 1, 0, 0, 1, 0, 0],
 [1, 1, 1, 0, 0, 0, 1, 0],
 [1, 1, 0, 1, 0, 1, 0, 0],
 [1, 1, 0, 1, 0, 0, 1, 0]]
sage: dw.tamari_interval(DyckWord([1,1,0,0,1,1,0,0]))
Traceback (most recent call last):
...
ValueError: the two Dyck words are not comparable on the Tamari lattice

to_area_sequence()¶

Return the area sequence of the Dyck word self.

The area sequence of a Dyck word \(w\) is defined as follows: Representing the Dyck word \(w\) as a Dyck path from \((0, 0)\) to \((n, n)\) using \(N\) and \(E\) steps (this involves padding \(w\) by \(E\) steps until \(w\) reaches the main diagonal if \(w\) is not already a complete Dyck path), the area sequence of \(w\) is the sequence \((a_1, a_2, \ldots, a_n)\), where \(a_i\) is the number of full cells in the \(i\)-th row of the rectangle \([0, n] \times [0, n]\) which lie completely above the diagonal. (The cells are the regions into which the rectangle is subdivided by the lines \(x = i\) with \(i\) integer and the lines \(y = j\) with \(j\) integer. The \(i\)-th row consists of all the cells between the lines \(y = i-1\) and \(y = i\).)

An alternative definition: Representing the Dyck word \(w\) as a Dyck path consisting of \(NE\) and \(SE\) steps, the area sequence is the sequence of ordinates of all lattice points on the path which are starting points of \(NE\) steps.

A list of integers \(l\) is the area sequence of some Dyck path if and only if it satisfies \(l_0 = 0\) and \(0 \leq l_{i+1} \leq l_i + 1\) for \(i > 0\).

EXAMPLES:

sage: DyckWord([]).to_area_sequence()
[]
sage: DyckWord([1, 0]).to_area_sequence()
[0]
sage: DyckWord([1, 1, 0, 0]).to_area_sequence()
[0, 1]
sage: DyckWord([1, 0, 1, 0]).to_area_sequence()
[0, 0]
sage: all(dw ==
....:     DyckWords().from_area_sequence(dw.to_area_sequence())
....:     for i in range(6) for dw in DyckWords(i))
True
sage: DyckWord([1,0,1,0,1,0,1,0,1,0]).to_area_sequence()
[0, 0, 0, 0, 0]
sage: DyckWord([1,1,1,1,1,0,0,0,0,0]).to_area_sequence()
[0, 1, 2, 3, 4]
sage: DyckWord([1,1,1,1,0,1,0,0,0,0]).to_area_sequence()
[0, 1, 2, 3, 3]
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_area_sequence()
[0, 1, 1, 0, 1, 1, 1]

to_binary_tree(usemap='1L0R')¶

Return a binary tree recursively constructed from the Dyck path self by the map usemap. The default usemap is '1L0R' which means:

an empty Dyck word is a leaf,
a non empty Dyck word reads \(1 L 0 R\) where \(L\) and \(R\) correspond to respectively its left and right subtrees.

INPUT:

usemap – a string, either '1L0R', '1R0L', 'L1R0', 'R1L0'

Other valid usemap are '1R0L', 'L1R0', and 'R1L0'. These correspond to different maps from Dyck paths to binary trees, whose recursive definitions are hopefully clear from the names.

EXAMPLES:

sage: dw = DyckWord([1,0])
sage: dw.to_binary_tree()
[., .]
sage: dw = DyckWord([])
sage: dw.to_binary_tree()
.
sage: dw = DyckWord([1,0,1,1,0,0])
sage: dw.to_binary_tree()
[., [[., .], .]]
sage: dw.to_binary_tree("L1R0")
[[., .], [., .]]
sage: dw = DyckWord([1,0,1,1,0,0,1,1,1,0,1,0,0,0])
sage: dw.to_binary_tree() == dw.to_binary_tree("1R0L").left_right_symmetry()
True
sage: dw.to_binary_tree() == dw.to_binary_tree("L1R0").left_border_symmetry()
False
sage: dw.to_binary_tree("1R0L") == dw.to_binary_tree("L1R0").left_border_symmetry()
True
sage: dw.to_binary_tree("R1L0") == dw.to_binary_tree("L1R0").left_right_symmetry()
True
sage: dw.to_binary_tree("R10L")
Traceback (most recent call last):
...
ValueError: R10L is not a correct map

to_binary_tree_tamari()¶

Return the binary tree corresponding to self in a way which is consistent with the Tamari orders on the set of Dyck paths and on the set of binary trees.

This is the 'L1R0' map documented in to_binary_tree().

EXAMPLES:

sage: DyckWord([1,0]).to_binary_tree_tamari()
[., .]
sage: DyckWord([1,0,1,1,0,0]).to_binary_tree_tamari()
[[., .], [., .]]
sage: DyckWord([1,0,1,0,1,0]).to_binary_tree_tamari()
[[[., .], .], .]

to_path_string(unicode=False)¶

Return a path representation of the Dyck word consisting of steps / and \ .

INPUT:

unicode – boolean (default False) whether to use unicode

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())
    /\
 /\/  \/\/\
/          \

to_standard_tableau()¶

Return a standard tableau of shape \((a,b)\) where \(a\) is the number of open symbols and \(b\) is the number of close symbols in self.

EXAMPLES:

sage: DyckWord([]).to_standard_tableau()
[]
sage: DyckWord([1, 0]).to_standard_tableau()
[[1], [2]]
sage: DyckWord([1, 1, 0, 0]).to_standard_tableau()
[[1, 2], [3, 4]]
sage: DyckWord([1, 0, 1, 0]).to_standard_tableau()
[[1, 3], [2, 4]]
sage: DyckWord([1]).to_standard_tableau()
[[1]]
sage: DyckWord([1, 0, 1]).to_standard_tableau()
[[1, 3], [2]]

to_tamari_sorting_tuple()¶

Convert a Dyck word to a Tamari sorting tuple.

The result is a list of integers, one for every up-step from left to right. To each up-step is associated the distance to the corresponding down step in the Dyck word.

This is useful for a faster conversion to binary trees.

EXAMPLES:

sage: DyckWord([]).to_tamari_sorting_tuple()
[]
sage: DyckWord([1, 0]).to_tamari_sorting_tuple()
[0]
sage: DyckWord([1, 1, 0, 0]).to_tamari_sorting_tuple()
[1, 0]
sage: DyckWord([1, 0, 1, 0]).to_tamari_sorting_tuple()
[0, 0]
sage: DyckWord([1, 1, 0, 1, 0, 0]).to_tamari_sorting_tuple()
[2, 0, 0]

See also

to_Catalan_code()

touch_composition()¶

Return a composition which indicates the positions where self returns to the diagonal.

This assumes self to be a complete Dyck word.

OUTPUT:

a composition of length equal to the length of the Dyck word.

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).touch_composition()
[1, 1]
sage: DyckWord([1, 1, 0, 0]).touch_composition()
[2]
sage: DyckWord([1, 1, 0, 0, 1, 0]).touch_composition()
[2, 1]
sage: DyckWord([1, 0, 1, 1, 0, 0]).touch_composition()
[1, 2]
sage: DyckWord([]).touch_composition()
[]

touch_points()¶

Return the abscissae (or, equivalently, ordinates) of the points where the Dyck path corresponding to self (comprising \(NE\) and \(SE\) steps) touches the main diagonal. This includes the last point (if it is on the main diagonal) but excludes the beginning point.

Note that these abscissae are precisely the entries of returns_to_zero() divided by \(2\).

OUTPUT:

a list of integers indicating where the path touches the diagonal

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).touch_points()
[1, 2]
sage: DyckWord([1, 1, 0, 0]).touch_points()
[2]
sage: DyckWord([1, 1, 0, 0, 1, 0]).touch_points()
[2, 3]
sage: DyckWord([1, 0, 1, 1, 0, 0]).touch_points()
[1, 3]

valleys()¶

Return a list of the positions of the valleys of a Dyck word.

A valley is \(0\) followed by a \(1\).

See also

peaks(), number_of_valleys()

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).valleys()
[1]
sage: DyckWord([1, 1, 0, 0]).valleys()
[]
sage: DyckWord([1,1,0,1,0,1,0,0]).valleys()
[2, 4]

class sage.combinat.dyck_word.DyckWordBacktracker(k1, k2)¶

Bases: sage.combinat.backtrack.GenericBacktracker

This class is an iterator for all Dyck words with \(n\) opening parentheses and \(n - k\) closing parentheses using the backtracker class. It is used by the DyckWords_size class.

This is not really meant to be called directly, partially because it fails in a couple corner cases: DWB(0) yields [0], not the empty word, and DWB(k, k+1) yields something (it shouldn’t yield anything). This could be fixed with a sanity check in _rec(), but then we’d be doing the sanity check every time we generate new objects; instead, we do one sanity check in DyckWords and assume here that the sanity check has already been made.

AUTHOR:

Dan Drake (2008-05-30)

class sage.combinat.dyck_word.DyckWord_complete(parent, l, latex_options={})¶

Bases: sage.combinat.dyck_word.DyckWord

The class of complete Dyck words. A Dyck word is complete, if it contains as many closers as openers.

For further information on Dyck words, see DyckWords_class.

area()¶

Return the area for self corresponding to the area of the Dyck path.

One can view a balanced Dyck word as a lattice path from \((0,0)\) to \((n,n)\) in the first quadrant by letting ‘1’s represent steps in the direction \((1,0)\) and ‘0’s represent steps in the direction \((0,1)\). The resulting path will remain weakly above the diagonal \(y = x\).

The area statistic is the number of complete squares in the integer lattice which are below the path and above the line \(y = x\). The ‘half-squares’ directly above the line \(y = x\) do not contribute to this statistic.

EXAMPLES:

sage: dw = DyckWord([1,0,1,0])
sage: dw.area() # 2 half-squares, 0 complete squares
0

sage: dw = DyckWord([1,1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0,0])
sage: dw.area()
19

sage: DyckWord([1,1,1,1,0,0,0,0]).area()
6
sage: DyckWord([1,1,1,0,1,0,0,0]).area()
5
sage: DyckWord([1,1,1,0,0,1,0,0]).area()
4
sage: DyckWord([1,1,1,0,0,0,1,0]).area()
3
sage: DyckWord([1,0,1,1,0,1,0,0]).area()
2
sage: DyckWord([1,1,0,1,1,0,0,0]).area()
4
sage: DyckWord([1,1,0,0,1,1,0,0]).area()
2
sage: DyckWord([1,0,1,1,1,0,0,0]).area()
3
sage: DyckWord([1,0,1,1,0,0,1,0]).area()
1
sage: DyckWord([1,0,1,0,1,1,0,0]).area()
1
sage: DyckWord([1,1,0,0,1,0,1,0]).area()
1
sage: DyckWord([1,1,0,1,0,0,1,0]).area()
2
sage: DyckWord([1,1,0,1,0,1,0,0]).area()
3
sage: DyckWord([1,0,1,0,1,0,1,0]).area()
0

area_dinv_to_bounce_area_map()¶

Return the image of self under the map which sends a Dyck word with area equal to \(r\) and dinv equal to \(s\) to a Dyck word with bounce equal to \(r\) and area equal to \(s\) .

The inverse of this map is bounce_area_to_area_dinv_map().

For a definition of this map, see [Hag2008] p. 50 where it is called \(\zeta\). However, this map differs from Haglund’s map by an application of reverse() (as does the definition of the bounce() statistic).

EXAMPLES:

sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area_dinv_to_bounce_area_map()
[1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0]
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
5
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv()
13
sage: DyckWord([1,1,1,1,1,0,0,0,1,0,0,1,0,0]).area()
13
sage: DyckWord([1,1,1,1,1,0,0,0,1,0,0,1,0,0]).bounce()
5
sage: DyckWord([1,1,1,1,1,0,0,0,1,0,0,1,0,0]).area_dinv_to_bounce_area_map()
[1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]
sage: DyckWord([1,1,0,0]).area_dinv_to_bounce_area_map()
[1, 0, 1, 0]
sage: DyckWord([1,0,1,0]).area_dinv_to_bounce_area_map()
[1, 1, 0, 0]

bounce()¶

Return the bounce statistic of self due to J. Haglund, see [Hag2008].

One can view a balanced Dyck word as a lattice path from \((0,0)\) to \((n,n)\) in the first quadrant by letting ‘1’s represent steps in the direction \((0,1)\) and ‘0’s represent steps in the direction \((1,0)\). The resulting path will remain weakly above the diagonal \(y = x\).

We describe the bounce statistic of such a path in terms of what is known as the “bounce path”.

We can think of our bounce path as describing the trail of a billiard ball shot West from \((n, n)\), which “bounces” down whenever it encounters a vertical step and “bounces” left when it encounters the line \(y = x\).

The bouncing ball will strike the diagonal at the places

\[(0, 0), (j_1, j_1), (j_2, j_2), \ldots, (j_r-1, j_r-1), (j_r, j_r) = (n, n).\]

We define the bounce to be the sum \(\sum_{i=1}^{r-1} j_i\).

EXAMPLES:

sage: DyckWord([1,1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0,0]).bounce()
7
sage: DyckWord([1,1,1,1,0,0,0,0]).bounce()
0
sage: DyckWord([1,1,1,0,1,0,0,0]).bounce()
1
sage: DyckWord([1,1,1,0,0,1,0,0]).bounce()
2
sage: DyckWord([1,1,1,0,0,0,1,0]).bounce()
3
sage: DyckWord([1,0,1,1,0,1,0,0]).bounce()
3
sage: DyckWord([1,1,0,1,1,0,0,0]).bounce()
1
sage: DyckWord([1,1,0,0,1,1,0,0]).bounce()
2
sage: DyckWord([1,0,1,1,1,0,0,0]).bounce()
1
sage: DyckWord([1,0,1,1,0,0,1,0]).bounce()
4
sage: DyckWord([1,0,1,0,1,1,0,0]).bounce()
3
sage: DyckWord([1,1,0,0,1,0,1,0]).bounce()
5
sage: DyckWord([1,1,0,1,0,0,1,0]).bounce()
4
sage: DyckWord([1,1,0,1,0,1,0,0]).bounce()
2
sage: DyckWord([1,0,1,0,1,0,1,0]).bounce()
6

bounce_area_to_area_dinv_map()¶

Return the image of the Dyck word under the map which sends a Dyck word with bounce equal to \(r\) and area equal to \(s\) to a Dyck word with area equal to \(r\) and dinv equal to \(s\) .

This implementation uses a recursive method by saying that the last entry in the area sequence of the Dyck word self is equal to the number of touch points of the Dyck path minus 1 of the image of this map.

The inverse of this map is area_dinv_to_bounce_area_map().

For a definition of this map, see [Hag2008] p. 50 where it is called \(\zeta^{-1}\). However, this map differs from Haglund’s map by an application of reverse() (as does the definition of the bounce() statistic).

EXAMPLES:

sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).bounce_area_to_area_dinv_map()
[1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0]
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
5
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).bounce()
9
sage: DyckWord([1,1,0,0,1,1,1,1,0,0,1,0,0,0]).area()
9
sage: DyckWord([1,1,0,0,1,1,1,1,0,0,1,0,0,0]).dinv()
5
sage: all(D==D.bounce_area_to_area_dinv_map().area_dinv_to_bounce_area_map() for D in DyckWords(6))
True
sage: DyckWord([1,1,0,0]).bounce_area_to_area_dinv_map()
[1, 0, 1, 0]
sage: DyckWord([1,0,1,0]).bounce_area_to_area_dinv_map()
[1, 1, 0, 0]

bounce_path()¶

Return the bounce path of self formed by starting at \((n,n)\) and traveling West until encountering the first vertical step of self, then South until encountering the diagonal, then West again to hit the path, etc. until the \((0,0)\) point is reached. The path followed by this walk is the bounce path.

See also

bounce()

EXAMPLES:

sage: DyckWord([1,1,0,1,0,0]).bounce_path()
[1, 0, 1, 1, 0, 0]
sage: DyckWord([1,1,1,0,0,0]).bounce_path()
[1, 1, 1, 0, 0, 0]
sage: DyckWord([1,0,1,0,1,0]).bounce_path()
[1, 0, 1, 0, 1, 0]
sage: DyckWord([1,1,1,1,0,0,1,0,0,0]).bounce_path()
[1, 1, 0, 0, 1, 1, 1, 0, 0, 0]

characteristic_symmetric_function(q=None, R=Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field)¶

The characteristic function of self is the sum of \(q^{dinv(D,F)} Q_{ides(read(D,F))}\) over all permutation fillings of the Dyck path representing self, where \(ides(read(D,F))\) is the descent composition of the inverse of the reading word of the filling.

INPUT:

q – (default: q = R('q')) a parameter for the generating function power
R – (default : R = QQ['q','t'].fraction_field()) the base ring to do the calculations over

OUTPUT:

an element of the symmetric functions over the ring R (in the Schur basis).

EXAMPLES:

sage: R = QQ['q','t'].fraction_field()
sage: (q,t) = R.gens()
sage: f = sum(t**D.area()*D.characteristic_symmetric_function() for D in DyckWords(3)); f
(q^3+q^2*t+q*t^2+t^3+q*t)*s[1, 1, 1] + (q^2+q*t+t^2+q+t)*s[2, 1] + s[3]
sage: f.nabla(power=-1)
s[1, 1, 1]

decomposition_reverse()¶

Return the involution of self with a recursive definition.

If a Dyck word \(D\) decomposes as \(1 D_1 0 D_2\) where \(D_1\) and \(D_2\) are complete Dyck words then the decomposition reverse is \(1 \phi(D_2) 0 \phi(D_1)\).

EXAMPLES:

sage: DyckWord([1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0]).decomposition_reverse()
[1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]
sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).decomposition_reverse()
[1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0]
sage: DyckWord([1,1,0,0]).decomposition_reverse()
[1, 0, 1, 0]
sage: DyckWord([1,0,1,0]).decomposition_reverse()
[1, 1, 0, 0]

dinv(labeling=None)¶

Return the dinv statistic of self due to M. Haiman, see [Hag2008].

If a labeling is provided then this function returns the dinv of the labeled Dyck word.

INPUT:

labeling – an optional argument to be viewed as the labelings of the vertical edges of the Dyck path

OUTPUT:

an integer representing the dinv statistic of the Dyck path or the labelled Dyck path.

EXAMPLES:

sage: DyckWord([1,0,1,0,1,0,1,0,1,0]).dinv()
10
sage: DyckWord([1,1,1,1,1,0,0,0,0,0]).dinv()
0
sage: DyckWord([1,1,1,1,0,1,0,0,0,0]).dinv()
1
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv()
13
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv([1,2,3,4,5,6,7])
11
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv([6,7,5,3,4,2,1])
2

first_return_decomposition()¶

Decompose a Dyck word into a pair of Dyck words (potentially empty) where the first word consists of the word after the first up step and the corresponding matching closing parenthesis.

EXAMPLES:

sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).first_return_decomposition()
([1, 0, 1, 0], [1, 1, 0, 1, 0, 1, 0, 0])
sage: DyckWord([1,1,0,0]).first_return_decomposition()
([1, 0], [])
sage: DyckWord([1,0,1,0]).first_return_decomposition()
([], [1, 0])

list_parking_functions()¶

Return all parking functions whose supporting Dyck path is self.

EXAMPLES:

sage: DyckWord([1,1,0,0,1,0]).list_parking_functions()
Permutations of the multi-set [1, 1, 3]
sage: DyckWord([1,1,1,0,0,0]).list_parking_functions()
Permutations of the multi-set [1, 1, 1]
sage: DyckWord([1,0,1,0,1,0]).list_parking_functions()
Standard permutations of 3

major_index()¶

Return the major index of self .

The major index of a Dyck word \(D\) is the sum of the positions of the valleys of \(D\) (when started counting at position 1).

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).major_index()
2
sage: DyckWord([1, 1, 0, 0]).major_index()
0
sage: DyckWord([1, 1, 0, 0, 1, 0]).major_index()
4
sage: DyckWord([1, 0, 1, 1, 0, 0]).major_index()
2

number_of_parking_functions()¶

Return the number of parking functions with self as the supporting Dyck path.

One representation of a parking function is as a pair consisting of a Dyck path and a permutation \(\pi\) such that if \([a_0, a_1, \ldots, a_{n-1}]\) is the area_sequence of the Dyck path (see to_area_sequence) then the permutation \(\pi\) satisfies \(\pi_i < \pi_{i+1}\) whenever \(a_{i} < a_{i+1}\). This function counts the number of permutations \(\pi\) which satisfy this condition.

EXAMPLES:

sage: DyckWord(area_sequence=[0,1,2]).number_of_parking_functions()
1
sage: DyckWord(area_sequence=[0,1,1]).number_of_parking_functions()
3
sage: DyckWord(area_sequence=[0,1,0]).number_of_parking_functions()
3
sage: DyckWord(area_sequence=[0,0,0]).number_of_parking_functions()
6

number_of_tunnels(tunnel_type='centered')¶

Return the number of tunnels of self of type tunnel_type.

A tunnel is a pair \((a,b)\) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If \(a + b = n\) then the tunnel is called centered . If \(a + b < n\) then the tunnel is called left and if \(a + b > n\), then the tunnel is called right.

INPUT:

tunnel_type – (default: 'centered') can be one of the following: 'left', 'right', 'centered', or 'all'.

EXAMPLES:

sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels()
0
sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels('left')
5
sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels('right')
2
sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels('all')
7
sage: DyckWord([1, 1, 0, 0]).number_of_tunnels('centered')
2

pyramid_weight()¶

Return the pyramid weight of self.

A pyramid of self is a subsequence of the form \(1^h 0^h\). A pyramid is maximal if it is neither preceded by a \(1\) nor followed by a \(0\).

The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids and was defined in [DS1992].

EXAMPLES:

sage: DyckWord([1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]).pyramid_weight()
6
sage: DyckWord([1,1,1,0,0,0]).pyramid_weight()
3
sage: DyckWord([1,0,1,0,1,0]).pyramid_weight()
3
sage: DyckWord([1,1,0,1,0,0]).pyramid_weight()
2

reading_permutation()¶

Return the reading permutation of self.

This is the permutation formed by taking the reading word of the Dyck path representing self (with \(N\) and \(E\) steps) if the vertical edges of the Dyck path are labeled from bottom to top with \(1\) through \(n\) and the diagonals are read from top to bottom starting with the diagonal furthest from the main diagonal.

EXAMPLES:

sage: DyckWord([1,0,1,0]).reading_permutation()
[2, 1]
sage: DyckWord([1,1,0,0]).reading_permutation()
[2, 1]
sage: DyckWord([1,1,0,1,0,0]).reading_permutation()
[3, 2, 1]
sage: DyckWord([1,1,0,0,1,0]).reading_permutation()
[2, 3, 1]
sage: DyckWord([1,0,1,1,0,0,1,0]).reading_permutation()
[3, 4, 2, 1]

reverse()¶

Return the reverse and complement of self.

This operation corresponds to flipping the Dyck path across the \(y=-x\) line.

EXAMPLES:

sage: DyckWord([1,1,0,0,1,0]).reverse()
[1, 0, 1, 1, 0, 0]
sage: DyckWord([1,1,1,0,0,0]).reverse()
[1, 1, 1, 0, 0, 0]
sage: len([D for D in DyckWords(5) if D.reverse() == D])
10

semilength()¶

Return the semilength of self.

The semilength of a complete Dyck word \(d\) is the number of openers and the number of closers.

EXAMPLES:

sage: DyckWord([1, 0, 1, 0]).semilength()
2

to_132_avoiding_permutation()¶

Use the bijection by C. Krattenthaler in [Kra2001] to send self to a \(132\)-avoiding permutation.

EXAMPLES:

sage: DyckWord([1,1,0,0]).to_132_avoiding_permutation()
[1, 2]
sage: DyckWord([1,0,1,0]).to_132_avoiding_permutation()
[2, 1]
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_132_avoiding_permutation()
[6, 5, 4, 7, 2, 1, 3]

to_312_avoiding_permutation()¶

Convert self to a \(312\)-avoiding permutation using the bijection by Bandlow and Killpatrick in [BK2001]. Sends the area to the inversion number.

EXAMPLES:

sage: DyckWord([1,1,0,0]).to_312_avoiding_permutation()
[2, 1]
sage: DyckWord([1,0,1,0]).to_312_avoiding_permutation()
[1, 2]
sage: p = DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_312_avoiding_permutation(); p
[2, 3, 1, 5, 6, 7, 4]
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
5
sage: p.length()
5

to_321_avoiding_permutation()¶

Use the bijection (pp. 60-61 of [Knu1973] or section 3.1 of [CK2008]) to send self to a \(321\)-avoiding permutation.

It is shown in [EP2004] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of excedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.

EXAMPLES:

sage: DyckWord([1,0,1,0]).to_321_avoiding_permutation()
[2, 1]
sage: DyckWord([1,1,0,0]).to_321_avoiding_permutation()
[1, 2]
sage: D = DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0])
sage: p = D.to_321_avoiding_permutation()
sage: p
[3, 5, 1, 6, 2, 7, 4]
sage: D.number_of_tunnels()
0
sage: p.number_of_fixed_points()
0
sage: D.number_of_tunnels('right')
4
sage: len(p.weak_excedences())-p.number_of_fixed_points()
4
sage: n = D.semilength()
sage: D.heights()[n] + n
8
sage: 2*p.longest_increasing_subsequence_length()
8

to_Catalan_code()¶

Return the Catalan code associated to self.

A Catalan code of length \(n\) is a sequence \((a_1, a_2, \ldots, a_n)\) of \(n\) integers \(a_i\) such that:

\(0 \leq a_i \leq n-i\) for every \(i\);
if \(i < j\) and \(a_i > 0\) and \(a_j > 0\) and \(a_{i+1} = a_{i+2} = \cdots = a_{j-1} = 0\), then \(a_i - a_j < j-i\).

It turns out that the Catalan codes of length \(n\) are in bijection with Dyck words.

The Catalan code of a Dyck word is example (x) in Richard Stanley’s exercises on combinatorial interpretations for Catalan objects. The code in this example is the reverse of the description provided there. See [Sta-EC2] and [StaCat98].

EXAMPLES:

sage: DyckWord([]).to_Catalan_code()
[]
sage: DyckWord([1, 0]).to_Catalan_code()
[0]
sage: DyckWord([1, 1, 0, 0]).to_Catalan_code()
[0, 1]
sage: DyckWord([1, 0, 1, 0]).to_Catalan_code()
[0, 0]
sage: all(dw ==
....:     DyckWords().from_Catalan_code(dw.to_Catalan_code())
....:     for i in range(6) for dw in DyckWords(i))
True

See also

to_tamari_sorting_tuple()

to_alternating_sign_matrix()¶

Return self as an alternating sign matrix.

This is an inclusion map from Dyck words of length \(2n\) to certain \(n \times n\) alternating sign matrices.

EXAMPLES:

sage: DyckWord([1,1,1,0,1,0,0,0]).to_alternating_sign_matrix()
[ 0  0  1  0]
[ 1  0 -1  1]
[ 0  1  0  0]
[ 0  0  1  0]
sage: DyckWord([1,0,1,0,1,1,0,0]).to_alternating_sign_matrix()
[1 0 0 0]
[0 1 0 0]
[0 0 0 1]
[0 0 1 0]

to_non_decreasing_parking_function()¶

Bijection to non-decreasing parking functions.

See there the method to_dyck_word() for more information.

EXAMPLES:

sage: DyckWord([]).to_non_decreasing_parking_function()
[]
sage: DyckWord([1,0]).to_non_decreasing_parking_function()
[1]
sage: DyckWord([1,1,0,0]).to_non_decreasing_parking_function()
[1, 1]
sage: DyckWord([1,0,1,0]).to_non_decreasing_parking_function()
[1, 2]
sage: DyckWord([1,0,1,1,0,1,0,0,1,0]).to_non_decreasing_parking_function()
[1, 2, 2, 3, 5]

to_noncrossing_partition(bijection=None)¶

Bijection of Biane from self to a noncrossing partition.

There is an optional parameter bijection that indicates if a different bijection from Dyck words to non-crossing partitions should be used (since there are potentially many).

If the parameter bijection is “Stump” then the bijection used is from [Stu2008], see also the method to_noncrossing_permutation().

Thanks to Mathieu Dutour for describing the bijection. See also from_noncrossing_partition().

EXAMPLES:

sage: DyckWord([]).to_noncrossing_partition()
[]
sage: DyckWord([1, 0]).to_noncrossing_partition()
[[1]]
sage: DyckWord([1, 1, 0, 0]).to_noncrossing_partition()
[[1, 2]]
sage: DyckWord([1, 1, 1, 0, 0, 0]).to_noncrossing_partition()
[[1, 2, 3]]
sage: DyckWord([1, 0, 1, 0, 1, 0]).to_noncrossing_partition()
[[1], [2], [3]]
sage: DyckWord([1, 1, 0, 1, 0, 0]).to_noncrossing_partition()
[[2], [1, 3]]
sage: DyckWord([]).to_noncrossing_partition("Stump")
[]
sage: DyckWord([1, 0]).to_noncrossing_partition("Stump")
[[1]]
sage: DyckWord([1, 1, 0, 0]).to_noncrossing_partition("Stump")
[[1, 2]]
sage: DyckWord([1, 1, 1, 0, 0, 0]).to_noncrossing_partition("Stump")
[[1, 3], [2]]
sage: DyckWord([1, 0, 1, 0, 1, 0]).to_noncrossing_partition("Stump")
[[1], [2], [3]]
sage: DyckWord([1, 1, 0, 1, 0, 0]).to_noncrossing_partition("Stump")
[[1, 2, 3]]

to_noncrossing_permutation()¶

Use the bijection by C. Stump in [Stu2008] to send self to a non-crossing permutation.

A non-crossing permutation when written in cyclic notation has cycles which are strictly increasing. Sends the area to the inversion number and self.major_index() to \(n(n-1) - maj(\sigma) - maj(\sigma^{-1})\). Uses the function pealing()

EXAMPLES:

sage: DyckWord([1,1,0,0]).to_noncrossing_permutation()
[2, 1]
sage: DyckWord([1,0,1,0]).to_noncrossing_permutation()
[1, 2]
sage: p = DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_noncrossing_permutation(); p
[2, 3, 1, 5, 6, 7, 4]
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
5
sage: p.length()
5

to_ordered_tree()¶

Return the ordered tree corresponding to self where the depth of the tree is the maximal height of self.

EXAMPLES:

sage: D = DyckWord([1,1,0,0])
sage: D.to_ordered_tree()
[[[]]]
sage: D = DyckWord([1,0,1,0])
sage: D.to_ordered_tree()
[[], []]
sage: D = DyckWord([1, 0, 1, 1, 0, 0])
sage: D.to_ordered_tree()
[[], [[]]]
sage: D = DyckWord([1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0])
sage: D.to_ordered_tree()
[[], [[], []], [[], [[]]]]

to_pair_of_standard_tableaux()¶

Convert self to a pair of standard tableaux of the same shape and of length less than or equal to two.

EXAMPLES:

sage: DyckWord([1,0,1,0]).to_pair_of_standard_tableaux()
([[1], [2]], [[1], [2]])
sage: DyckWord([1,1,0,0]).to_pair_of_standard_tableaux()
([[1, 2]], [[1, 2]])
sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_pair_of_standard_tableaux()
([[1, 2, 4, 7], [3, 5, 6]], [[1, 2, 4, 6], [3, 5, 7]])

to_partition()¶

Return the partition associated to self .

This partition is determined by thinking of self as a lattice path and considering the cells which are above the path but within the \(n \times n\) grid and the partition is formed by reading the sequence of the number of cells in this collection in each row.

OUTPUT:

a partition representing the rows of cells in the square lattice and above the path

EXAMPLES:

sage: DyckWord([]).to_partition()
[]
sage: DyckWord([1,0]).to_partition()
[]
sage: DyckWord([1,1,0,0]).to_partition()
[]
sage: DyckWord([1,0,1,0]).to_partition()
[1]
sage: DyckWord([1,0,1,0,1,0]).to_partition()
[2, 1]
sage: DyckWord([1,1,0,0,1,0]).to_partition()
[2]
sage: DyckWord([1,0,1,1,0,0]).to_partition()
[1, 1]

to_permutation(map)¶

This is simply a method collecting all implemented maps from Dyck words to permutations.

INPUT:

map – defines the map from Dyck words to permutations. These are currently:
- Bandlow-Killpatrick: to_312_avoiding_permutation()
- Knuth: to_321_avoiding_permutation()
- Krattenthaler: to_132_avoiding_permutation()
- Stump: to_noncrossing_permutation()

EXAMPLES:

sage: D = DyckWord([1,1,1,0,1,0,0,0])
sage: D.pretty_print()
   _____
 _| x x
| x x  .
| x  . .
|  . . .

sage: D.to_permutation(map="Bandlow-Killpatrick")
[3, 4, 2, 1]
sage: D.to_permutation(map="Stump")
[4, 2, 3, 1]
sage: D.to_permutation(map="Knuth")
[1, 2, 4, 3]
sage: D.to_permutation(map="Krattenthaler")
[2, 1, 3, 4]

to_triangulation()¶

Map self to a triangulation.

The map from complete Dyck words of length \(2n\) to triangulations of \(n+2\)-gon given by this function is a bijection that can be described as follows.

Consider the Dyck word as a path from \((0, 0)\) to \((n, n)\) staying above the diagonal, where \(1\) is an up step and \(0\) is a right step. Then each horizontal step has a co-height (\(0\) at the top and \(n-1\) at most at the bottom). One reads the Dyck word from left to right. At the beginning, all vertices from \(0\) to \(n+1\) are available. For each horizontal step, one creates an edge from the vertex indexed by the co-height to the next available vertex. This chops out a triangle from the polygon and one removes the middle vertex of this triangle from the list of available vertices.

This bijection has the property that the set of smallest vertices of the edges in a triangulation is an encoding of the co-heights, from which the Dyck word can be easily recovered.

OUTPUT:

a list of pairs \((i, j)\) that are the edges of the triangulations.

EXAMPLES:

sage: DyckWord([1, 1, 0, 0]).to_triangulation()
[(0, 2)]

sage: [t.to_triangulation() for t in DyckWords(3)]
[[(2, 4), (1, 4)],
[(2, 4), (0, 2)],
[(1, 3), (1, 4)],
[(1, 3), (0, 3)],
[(0, 2), (0, 3)]]

REFERENCES:

[Cha2005]

to_triangulation_as_graph()¶

Map self to a triangulation and return the result as a graph.

See to_triangulation() for the bijection used to map complete Dyck words to triangulations.

OUTPUT:

a graph containing both the perimeter edges and the inner edges of a triangulation of a regular polygon.

EXAMPLES:

sage: g = DyckWord([1, 1, 0, 0, 1, 0]).to_triangulation_as_graph()
sage: g
Graph on 5 vertices
sage: g.edges(labels=False)
[(0, 1), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (3, 4)]
sage: g.show()        # not tested

tunnels()¶

Return an iterator of ranges of the matching parentheses in the Dyck word self.

That is, if (a,b) is in self.tunnels(), then the matching parenthesis to self[a] is self[b-1].

EXAMPLES:

sage: list(DyckWord([1, 1, 0, 1, 1, 0, 0, 1, 0, 0]).tunnels())
[(0, 10), (1, 3), (3, 7), (4, 6), (7, 9)]

class sage.combinat.dyck_word.DyckWords¶

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

Dyck words.

A Dyck word is a sequence \((w_1, \ldots, w_n)\) consisting of 1 s and 0 s, with the property that for any \(i\) with \(1 \leq i \leq n\), the sequence \((w_1, \ldots, w_i)\) contains at least as many 1 s as 0 s.

A Dyck word is balanced if the total number of 1 s is equal to the total number of 0 s. The number of balanced Dyck words of length \(2k\) is given by the Catalan number \(C_k\).

EXAMPLES:

This class can be called with three keyword parameters k1, k2 and complete.

If neither k1 nor k2 are specified, then DyckWords returns the combinatorial class of all balanced (=complete) Dyck words, unless the keyword complete is set to False (in which case it returns the class of all Dyck words).

sage: DW = DyckWords(); DW
Complete Dyck words
sage: [] in DW
True
sage: [1, 0, 1, 0] in DW
True
sage: [1, 1, 0] in DW
False
sage: ADW = DyckWords(complete=False); ADW
Dyck words
sage: [] in ADW
True
sage: [1, 0, 1, 0] in ADW
True
sage: [1, 1, 0] in ADW
True
sage: [1, 0, 0] in ADW
False

If just k1 is specified, then it returns the balanced Dyck words with k1 opening parentheses and k1 closing parentheses.

sage: DW2 = DyckWords(2); DW2
Dyck words with 2 opening parentheses and 2 closing parentheses
sage: DW2.first()
[1, 0, 1, 0]
sage: DW2.last()
[1, 1, 0, 0]
sage: DW2.cardinality()
2
sage: DyckWords(100).cardinality() == catalan_number(100)
True

If k2 is specified in addition to k1, then it returns the Dyck words with k1 opening parentheses and k2 closing parentheses.

sage: DW32 = DyckWords(3,2); DW32
Dyck words with 3 opening parentheses and 2 closing parentheses
sage: DW32.list()
[[1, 0, 1, 0, 1],
 [1, 0, 1, 1, 0],
 [1, 1, 0, 0, 1],
 [1, 1, 0, 1, 0],
 [1, 1, 1, 0, 0]]

Element¶: alias of DyckWord

from_heights(heights)¶

Compute a Dyck word knowing its heights.

We view the Dyck word as a Dyck path from \((0, 0)\) to \((2n, 0)\) in the first quadrant by letting 1’s represent steps in the direction \((1, 1)\) and 0’s represent steps in the direction \((1, -1)\).

The heights() is the sequence of the \(y\)-coordinates of the \(2n+1\) lattice points along this path.

EXAMPLES:

sage: from sage.combinat.dyck_word import DyckWord
sage: D = DyckWords(complete=False)
sage: D.from_heights((0,))
[]
sage: D.from_heights((0, 1, 0))
[1, 0]
sage: D.from_heights((0, 1, 2, 1, 0))
[1, 1, 0, 0]

This also works for incomplete Dyck words:

sage: D.from_heights((0, 1, 2, 1, 2, 1))
[1, 1, 0, 1, 0]
sage: D.from_heights((0, 1, 2, 1))
[1, 1, 0]

See also

heights(), min_from_heights()

min_from_heights(heights)¶

Compute the smallest Dyck word which achieves or surpasses a given sequence of heights.

INPUT:

heights – a nonempty list or iterable consisting of nonnegative integers, the first of which is \(0\)

OUTPUT:

The smallest Dyck word whose sequence of heights is componentwise higher-or-equal to heights. Here, “smaller” can be understood both in the sense of lexicographic order on the Dyck words, and in the sense of every vertex of the path having the smallest possible height.

See also

heights()
from_heights()

EXAMPLES:

sage: D = DyckWords(complete=False)
sage: D.min_from_heights((0,))
[]
sage: D.min_from_heights((0, 1, 0))
[1, 0]
sage: D.min_from_heights((0, 0, 2, 0, 0))
[1, 1, 0, 0]
sage: D.min_from_heights((0, 0, 2, 0, 2, 0))
[1, 1, 0, 1, 0]
sage: D.min_from_heights((0, 0, 1, 0, 1, 0))
[1, 1, 0, 1, 0]

options(*get_value, **set_value)¶

Set and display the 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 DyckWords.options() of DyckWords and related parent classes.

@OPTIONS

EXAMPLES:

sage: D = DyckWord([1, 1, 0, 1, 0, 0])
sage: D
[1, 1, 0, 1, 0, 0]
sage: DyckWords.options.display="lattice"
sage: D
   ___
 _| x
| x  .
|  . .
sage: DyckWords.options(diagram_style="line")
sage: D
 /\/\
/    \
sage: DyckWords.options._reset()

See GlobalOptions for more features of these options.

class sage.combinat.dyck_word.DyckWords_all¶

Bases: sage.combinat.dyck_word.DyckWords

All Dyck words.

class sage.combinat.dyck_word.DyckWords_size(k1, k2)¶

Bases: sage.combinat.dyck_word.DyckWords

Dyck words with \(k_1\) openers and \(k_2\) closers.

cardinality()¶

Return the number of Dyck words with \(k_1\) openers and \(k_2\) closers.

This number is

\[\frac{k_1 - k_2 + 1}{k_1 + 1} \binom{k_1 + k_2}{k_2} = \binom{k_1 + k_2}{k_2} - \binom{k_1 + k_2}{k_2 - 1}\]

if \(k_2 \leq k_1 + 1\), and \(0\) if \(k_2 > k_1\) (these numbers are the same if \(k_2 = k_1 + 1\)).

EXAMPLES:

sage: DyckWords(3, 2).cardinality()
5
sage: all(all(DyckWords(p, q).cardinality()
....:           == len(DyckWords(p, q).list()) for q in range(p + 1))
....:      for p in range(7))
True

sage.combinat.dyck_word.is_a(obj, k1=None, k2=None)¶

Test if obj is a Dyck word with exactly k1 open symbols and exactly k2 close symbols.

If k1 is not specified, then the number of open symbols can be arbitrary. If k1 is specified but k2 is not, then k2 is set to be k1.

EXAMPLES:

sage: from sage.combinat.dyck_word import is_a
sage: is_a([1,1,0,0])
True
sage: is_a([1,0,1,0])
True
sage: is_a([1,1,0,0], 2)
True
sage: is_a([1,1,0,0], 3)
False
sage: is_a([1,1,0,0], 3, 2)
False
sage: is_a([1,1,0])
True
sage: is_a([0,1,0])
False
sage: is_a([1,0,0])
False
sage: is_a([1,1,0],2,1)
True
sage: is_a([1,1,0],2)
False
sage: is_a([1,1,0],1,1)
False

sage.combinat.dyck_word.is_area_sequence(seq)¶

Test if a sequence \(l\) of integers satisfies \(l_0 = 0\) and \(0 \leq l_{i+1} \leq l_i + 1\) for \(i > 0\).

EXAMPLES:

sage: from sage.combinat.dyck_word import is_area_sequence
sage: is_area_sequence([0,2,0])
False
sage: is_area_sequence([1,2,3])
False
sage: is_area_sequence([0,1,0])
True
sage: is_area_sequence([0,1,2])
True
sage: is_area_sequence([])
True

sage.combinat.dyck_word.pealing(D, return_touches=False)¶

A helper function for computing the bijection from a Dyck word to a \(231\)-avoiding permutation using the bijection “Stump”. For details see [Stu2008].

See also

to_noncrossing_partition()

EXAMPLES:

sage: from sage.combinat.dyck_word import pealing
sage: pealing(DyckWord([1,1,0,0]))
[1, 0, 1, 0]
sage: pealing(DyckWord([1,0,1,0]))
[1, 0, 1, 0]
sage: pealing(DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]))
[1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
sage: pealing(DyckWord([1,1,0,0]),return_touches=True)
([1, 0, 1, 0], [[1, 2]])
sage: pealing(DyckWord([1,0,1,0]),return_touches=True)
([1, 0, 1, 0], [])
sage: pealing(DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]),return_touches=True)
([1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [[1, 2], [3, 5]])

sage.combinat.dyck_word.replace_parens(x)¶

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 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.

See also

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

sage.combinat.dyck_word.replace_symbols(x)¶

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 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.

See also

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

Dyck Words¶

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