Non-symmetric Macdonald Polynomials#

class sage.combinat.sf.ns_macdonald.AugmentedLatticeDiagramFilling(l, pi=None)[source]#

Bases: CombinatorialObject

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a == loads(dumps(a))
True
sage: pi = Permutation([2,3,1]).to_permutation_group_element()
sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]],pi)
sage: a == loads(dumps(a))
True

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a == loads(dumps(a))
True
>>> pi = Permutation([Integer(2),Integer(3),Integer(1)]).to_permutation_group_element()
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]],pi)
>>> a == loads(dumps(a))
True
are_attacking(i, j, ii, jj)[source]#

Return True if the boxes (i,j) and (ii,jj) in self are attacking.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: all( a.are_attacking(i,j,ii,jj) for (i,j),(ii,jj) in a.attacking_boxes())
True
sage: a.are_attacking(1,1,3,2)
False

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> all( a.are_attacking(i,j,ii,jj) for (i,j),(ii,jj) in a.attacking_boxes())
True
>>> a.are_attacking(Integer(1),Integer(1),Integer(3),Integer(2))
False
attacking_boxes()[source]#

Return a list of pairs of boxes in self that are attacking.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.attacking_boxes()[:5]
[((1, 1), (2, 1)),
 ((1, 1), (3, 1)),
 ((1, 1), (6, 1)),
 ((1, 1), (2, 0)),
 ((1, 1), (3, 0))]

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.attacking_boxes()[:Integer(5)]
[((1, 1), (2, 1)),
 ((1, 1), (3, 1)),
 ((1, 1), (6, 1)),
 ((1, 1), (2, 0)),
 ((1, 1), (3, 0))]
boxes()[source]#

Return a list of the coordinates of the boxes of self, including the ‘basement row’.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.boxes()
[(1, 1),
 (1, 2),
 (2, 1),
 (3, 1),
 (3, 2),
 (3, 3),
 (6, 1),
 (6, 2),
 (1, 0),
 (2, 0),
 (3, 0),
 (4, 0),
 (5, 0),
 (6, 0)]

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.boxes()
[(1, 1),
 (1, 2),
 (2, 1),
 (3, 1),
 (3, 2),
 (3, 3),
 (6, 1),
 (6, 2),
 (1, 0),
 (2, 0),
 (3, 0),
 (4, 0),
 (5, 0),
 (6, 0)]
coeff(q, t)[source]#

Return the coefficient in front of self in the HHL formula for the expansion of the non-symmetric Macdonald polynomial E(self.shape()).

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: q,t = var('q,t')                                                      # needs sage.symbolic
sage: a.coeff(q,t)                                                          # needs sage.symbolic
(t - 1)^4/((q^2*t^3 - 1)^2*(q*t^2 - 1)^2)

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> q,t = var('q,t')                                                      # needs sage.symbolic
>>> a.coeff(q,t)                                                          # needs sage.symbolic
(t - 1)^4/((q^2*t^3 - 1)^2*(q*t^2 - 1)^2)
coeff_integral(q, t)[source]#

Return the coefficient in front of self in the HHL formula for the expansion of the integral non-symmetric Macdonald polynomial E(self.shape())

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: q,t = var('q,t')                                                      # needs sage.symbolic
sage: a.coeff_integral(q,t)                                                 # needs sage.symbolic
(q^2*t^3 - 1)^2*(q*t^2 - 1)^2*(t - 1)^4

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> q,t = var('q,t')                                                      # needs sage.symbolic
>>> a.coeff_integral(q,t)                                                 # needs sage.symbolic
(q^2*t^3 - 1)^2*(q*t^2 - 1)^2*(t - 1)^4
coinv()[source]#

Return self’s co-inversion statistic.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.coinv()
2

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.coinv()
2
descents()[source]#

Return a list of the descents of self.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.descents()
[(1, 2), (3, 2)]

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.descents()
[(1, 2), (3, 2)]
inv()[source]#

Return self’s inversion statistic.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.inv()
15

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.inv()
15
inversions()[source]#

Return a list of the inversions of self.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.inversions()[:5]
[((6, 2), (3, 2)),
 ((1, 2), (6, 1)),
 ((1, 2), (3, 1)),
 ((1, 2), (2, 1)),
 ((6, 1), (3, 1))]
sage: len(a.inversions())
25

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.inversions()[:Integer(5)]
[((6, 2), (3, 2)),
 ((1, 2), (6, 1)),
 ((1, 2), (3, 1)),
 ((1, 2), (2, 1)),
 ((6, 1), (3, 1))]
>>> len(a.inversions())
25
is_non_attacking()[source]#

Return True if self is non-attacking.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.is_non_attacking()
True
sage: a = AugmentedLatticeDiagramFilling([[1, 1, 1], [2, 3], [3]])
sage: a.is_non_attacking()
False
sage: a = AugmentedLatticeDiagramFilling([[2,2],[1]])
sage: a.is_non_attacking()
False
sage: pi = Permutation([2,1]).to_permutation_group_element()
sage: a = AugmentedLatticeDiagramFilling([[2,2],[1]],pi)
sage: a.is_non_attacking()
True

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.is_non_attacking()
True
>>> a = AugmentedLatticeDiagramFilling([[Integer(1), Integer(1), Integer(1)], [Integer(2), Integer(3)], [Integer(3)]])
>>> a.is_non_attacking()
False
>>> a = AugmentedLatticeDiagramFilling([[Integer(2),Integer(2)],[Integer(1)]])
>>> a.is_non_attacking()
False
>>> pi = Permutation([Integer(2),Integer(1)]).to_permutation_group_element()
>>> a = AugmentedLatticeDiagramFilling([[Integer(2),Integer(2)],[Integer(1)]],pi)
>>> a.is_non_attacking()
True
maj()[source]#

Return the major index of self.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.maj()
3

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.maj()
3
permuted_filling(sigma)[source]#

EXAMPLES:

sage: pi=Permutation([2,1,4,3]).to_permutation_group_element()
sage: fill=[[2],[1,2,3],[],[3,1]]
sage: AugmentedLatticeDiagramFilling(fill).permuted_filling(pi)
[[2, 1], [1, 2, 1, 4], [4], [3, 4, 2]]

>>> from sage.all import *
>>> pi=Permutation([Integer(2),Integer(1),Integer(4),Integer(3)]).to_permutation_group_element()
>>> fill=[[Integer(2)],[Integer(1),Integer(2),Integer(3)],[],[Integer(3),Integer(1)]]
>>> AugmentedLatticeDiagramFilling(fill).permuted_filling(pi)
[[2, 1], [1, 2, 1, 4], [4], [3, 4, 2]]
reading_order()[source]#

Return a list of coordinates of the boxes in self, starting from the top right, and reading from right to left.

Note that this includes the ‘basement row’ of self.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.reading_order()
[(3, 3),
 (6, 2),
 (3, 2),
 (1, 2),
 (6, 1),
 (3, 1),
 (2, 1),
 (1, 1),
 (6, 0),
 (5, 0),
 (4, 0),
 (3, 0),
 (2, 0),
 (1, 0)]

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.reading_order()
[(3, 3),
 (6, 2),
 (3, 2),
 (1, 2),
 (6, 1),
 (3, 1),
 (2, 1),
 (1, 1),
 (6, 0),
 (5, 0),
 (4, 0),
 (3, 0),
 (2, 0),
 (1, 0)]
reading_word()[source]#

Return the reading word of self, obtained by reading the boxes entries of self from right to left, starting in the upper right.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.reading_word()
word: 25465321

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.reading_word()
word: 25465321
shape()[source]#

Return the shape of self.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.shape()
[2, 1, 3, 0, 0, 2]

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.shape()
[2, 1, 3, 0, 0, 2]
weight()[source]#

Return the weight of self.

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.weight()
[1, 2, 1, 1, 2, 1]

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a.weight()
[1, 2, 1, 1, 2, 1]
sage.combinat.sf.ns_macdonald.E(mu, q=None, t=None, pi=None)[source]#

Return the non-symmetric Macdonald polynomial in type A corresponding to a shape mu, with basement permuted according to pi.

Note that if both \(q\) and \(t\) are specified, then they must have the same parent.

REFERENCE:

  • J. Haglund, M. Haiman, N. Loehr. A combinatorial formula for non-symmetric Macdonald polynomials. arXiv math/0601693v3.

See also

NonSymmetricMacdonaldPolynomials for a type free implementation where the polynomials are constructed recursively by the application of intertwining operators.

EXAMPLES:

sage: from sage.combinat.sf.ns_macdonald import E
sage: E([0,0,0])
1
sage: E([1,0,0])
x0
sage: E([0,1,0])
(t - 1)/(q*t^2 - 1)*x0 + x1
sage: E([0,0,1])
(t - 1)/(q*t - 1)*x0 + (t - 1)/(q*t - 1)*x1 + x2
sage: E([1,1,0])
x0*x1
sage: E([1,0,1])
(t - 1)/(q*t^2 - 1)*x0*x1 + x0*x2
sage: E([0,1,1])
(t - 1)/(q*t - 1)*x0*x1 + (t - 1)/(q*t - 1)*x0*x2 + x1*x2
sage: E([2,0,0])
x0^2 + (q*t - q)/(q*t - 1)*x0*x1 + (q*t - q)/(q*t - 1)*x0*x2
sage: E([0,2,0])
(t - 1)/(q^2*t^2 - 1)*x0^2 + (q^2*t^3 - q^2*t^2 + q*t^2 - 2*q*t + q - t + 1)/(q^3*t^3 - q^2*t^2 - q*t + 1)*x0*x1 + x1^2 + (q*t^2 - 2*q*t + q)/(q^3*t^3 - q^2*t^2 - q*t + 1)*x0*x2 + (q*t - q)/(q*t - 1)*x1*x2

>>> from sage.all import *
>>> from sage.combinat.sf.ns_macdonald import E
>>> E([Integer(0),Integer(0),Integer(0)])
1
>>> E([Integer(1),Integer(0),Integer(0)])
x0
>>> E([Integer(0),Integer(1),Integer(0)])
(t - 1)/(q*t^2 - 1)*x0 + x1
>>> E([Integer(0),Integer(0),Integer(1)])
(t - 1)/(q*t - 1)*x0 + (t - 1)/(q*t - 1)*x1 + x2
>>> E([Integer(1),Integer(1),Integer(0)])
x0*x1
>>> E([Integer(1),Integer(0),Integer(1)])
(t - 1)/(q*t^2 - 1)*x0*x1 + x0*x2
>>> E([Integer(0),Integer(1),Integer(1)])
(t - 1)/(q*t - 1)*x0*x1 + (t - 1)/(q*t - 1)*x0*x2 + x1*x2
>>> E([Integer(2),Integer(0),Integer(0)])
x0^2 + (q*t - q)/(q*t - 1)*x0*x1 + (q*t - q)/(q*t - 1)*x0*x2
>>> E([Integer(0),Integer(2),Integer(0)])
(t - 1)/(q^2*t^2 - 1)*x0^2 + (q^2*t^3 - q^2*t^2 + q*t^2 - 2*q*t + q - t + 1)/(q^3*t^3 - q^2*t^2 - q*t + 1)*x0*x1 + x1^2 + (q*t^2 - 2*q*t + q)/(q^3*t^3 - q^2*t^2 - q*t + 1)*x0*x2 + (q*t - q)/(q*t - 1)*x1*x2
sage.combinat.sf.ns_macdonald.E_integral(mu, q=None, t=None, pi=None)[source]#

Return the integral form for the non-symmetric Macdonald polynomial in type A corresponding to a shape mu.

Note that if both q and t are specified, then they must have the same parent.

REFERENCE:

  • J. Haglund, M. Haiman, N. Loehr. A combinatorial formula for non-symmetric Macdonald polynomials. arXiv math/0601693v3.

EXAMPLES:

sage: from sage.combinat.sf.ns_macdonald import E_integral
sage: E_integral([0,0,0])
1
sage: E_integral([1,0,0])
(-t + 1)*x0
sage: E_integral([0,1,0])
(-q*t^2 + 1)*x0 + (-t + 1)*x1
sage: E_integral([0,0,1])
(-q*t + 1)*x0 + (-q*t + 1)*x1 + (-t + 1)*x2
sage: E_integral([1,1,0])
(t^2 - 2*t + 1)*x0*x1
sage: E_integral([1,0,1])
(q*t^3 - q*t^2 - t + 1)*x0*x1 + (t^2 - 2*t + 1)*x0*x2
sage: E_integral([0,1,1])
(q^2*t^3 + q*t^4 - q*t^3 - q*t^2 - q*t - t^2 + t + 1)*x0*x1 + (q*t^2 - q*t - t + 1)*x0*x2 + (t^2 - 2*t + 1)*x1*x2
sage: E_integral([2,0,0])
(t^2 - 2*t + 1)*x0^2 + (q^2*t^2 - q^2*t - q*t + q)*x0*x1 + (q^2*t^2 - q^2*t - q*t + q)*x0*x2
sage: E_integral([0,2,0])
(q^2*t^3 - q^2*t^2 - t + 1)*x0^2 + (q^4*t^3 - q^3*t^2 - q^2*t + q*t^2 - q*t + q - t + 1)*x0*x1 + (t^2 - 2*t + 1)*x1^2 + (q^4*t^3 - q^3*t^2 - q^2*t + q)*x0*x2 + (q^2*t^2 - q^2*t - q*t + q)*x1*x2

>>> from sage.all import *
>>> from sage.combinat.sf.ns_macdonald import E_integral
>>> E_integral([Integer(0),Integer(0),Integer(0)])
1
>>> E_integral([Integer(1),Integer(0),Integer(0)])
(-t + 1)*x0
>>> E_integral([Integer(0),Integer(1),Integer(0)])
(-q*t^2 + 1)*x0 + (-t + 1)*x1
>>> E_integral([Integer(0),Integer(0),Integer(1)])
(-q*t + 1)*x0 + (-q*t + 1)*x1 + (-t + 1)*x2
>>> E_integral([Integer(1),Integer(1),Integer(0)])
(t^2 - 2*t + 1)*x0*x1
>>> E_integral([Integer(1),Integer(0),Integer(1)])
(q*t^3 - q*t^2 - t + 1)*x0*x1 + (t^2 - 2*t + 1)*x0*x2
>>> E_integral([Integer(0),Integer(1),Integer(1)])
(q^2*t^3 + q*t^4 - q*t^3 - q*t^2 - q*t - t^2 + t + 1)*x0*x1 + (q*t^2 - q*t - t + 1)*x0*x2 + (t^2 - 2*t + 1)*x1*x2
>>> E_integral([Integer(2),Integer(0),Integer(0)])
(t^2 - 2*t + 1)*x0^2 + (q^2*t^2 - q^2*t - q*t + q)*x0*x1 + (q^2*t^2 - q^2*t - q*t + q)*x0*x2
>>> E_integral([Integer(0),Integer(2),Integer(0)])
(q^2*t^3 - q^2*t^2 - t + 1)*x0^2 + (q^4*t^3 - q^3*t^2 - q^2*t + q*t^2 - q*t + q - t + 1)*x0*x1 + (t^2 - 2*t + 1)*x1^2 + (q^4*t^3 - q^3*t^2 - q^2*t + q)*x0*x2 + (q^2*t^2 - q^2*t - q*t + q)*x1*x2
sage.combinat.sf.ns_macdonald.Ht(mu, q=None, t=None, pi=None)[source]#

Return the symmetric Macdonald polynomial using the Haiman, Haglund, and Loehr formula.

Note that if both \(q\) and \(t\) are specified, then they must have the same parent.

REFERENCE:

  • J. Haglund, M. Haiman, N. Loehr. A combinatorial formula for non-symmetric Macdonald polynomials. arXiv math/0601693v3.

EXAMPLES:

sage: from sage.combinat.sf.ns_macdonald import Ht
sage: HHt = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().Ht()
sage: Ht([0,0,1])
x0 + x1 + x2
sage: HHt([1]).expand(3)
x0 + x1 + x2
sage: Ht([0,0,2])
x0^2 + (q + 1)*x0*x1 + x1^2 + (q + 1)*x0*x2 + (q + 1)*x1*x2 + x2^2
sage: HHt([2]).expand(3)
x0^2 + (q + 1)*x0*x1 + x1^2 + (q + 1)*x0*x2 + (q + 1)*x1*x2 + x2^2

>>> from sage.all import *
>>> from sage.combinat.sf.ns_macdonald import Ht
>>> HHt = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().Ht()
>>> Ht([Integer(0),Integer(0),Integer(1)])
x0 + x1 + x2
>>> HHt([Integer(1)]).expand(Integer(3))
x0 + x1 + x2
>>> Ht([Integer(0),Integer(0),Integer(2)])
x0^2 + (q + 1)*x0*x1 + x1^2 + (q + 1)*x0*x2 + (q + 1)*x1*x2 + x2^2
>>> HHt([Integer(2)]).expand(Integer(3))
x0^2 + (q + 1)*x0*x1 + x1^2 + (q + 1)*x0*x2 + (q + 1)*x1*x2 + x2^2
class sage.combinat.sf.ns_macdonald.LatticeDiagram(l, copy=True)[source]#

Bases: CombinatorialObject

a(i, j)[source]#

Return the length of the arm of the box (i,j) in self.

EXAMPLES:

sage: a = LatticeDiagram([3,1,2,4,3,0,4,2,3])
sage: a.a(5,2)
3

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(1),Integer(2),Integer(4),Integer(3),Integer(0),Integer(4),Integer(2),Integer(3)])
>>> a.a(Integer(5),Integer(2))
3
arm(i, j)[source]#

Return the arm of the box (i,j) in self.

EXAMPLES:

sage: a = LatticeDiagram([3,1,2,4,3,0,4,2,3])
sage: a.arm(5,2)
[(1, 2), (3, 2), (8, 1)]

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(1),Integer(2),Integer(4),Integer(3),Integer(0),Integer(4),Integer(2),Integer(3)])
>>> a.arm(Integer(5),Integer(2))
[(1, 2), (3, 2), (8, 1)]
arm_left(i, j)[source]#

Return the left arm of the box (i,j) in self.

EXAMPLES:

sage: a = LatticeDiagram([3,1,2,4,3,0,4,2,3])
sage: a.arm_left(5,2)
[(1, 2), (3, 2)]

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(1),Integer(2),Integer(4),Integer(3),Integer(0),Integer(4),Integer(2),Integer(3)])
>>> a.arm_left(Integer(5),Integer(2))
[(1, 2), (3, 2)]
arm_right(i, j)[source]#

Return the right arm of the box (i,j) in self.

EXAMPLES:

sage: a = LatticeDiagram([3,1,2,4,3,0,4,2,3])
sage: a.arm_right(5,2)
[(8, 1)]

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(1),Integer(2),Integer(4),Integer(3),Integer(0),Integer(4),Integer(2),Integer(3)])
>>> a.arm_right(Integer(5),Integer(2))
[(8, 1)]
boxes()[source]#

EXAMPLES:

sage: a = LatticeDiagram([3,0,2])
sage: a.boxes()
[(1, 1), (1, 2), (1, 3), (3, 1), (3, 2)]
sage: a = LatticeDiagram([2, 1, 3, 0, 0, 2])
sage: a.boxes()
[(1, 1), (1, 2), (2, 1), (3, 1), (3, 2), (3, 3), (6, 1), (6, 2)]

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(0),Integer(2)])
>>> a.boxes()
[(1, 1), (1, 2), (1, 3), (3, 1), (3, 2)]
>>> a = LatticeDiagram([Integer(2), Integer(1), Integer(3), Integer(0), Integer(0), Integer(2)])
>>> a.boxes()
[(1, 1), (1, 2), (2, 1), (3, 1), (3, 2), (3, 3), (6, 1), (6, 2)]
boxes_same_and_lower_right(ii, jj)[source]#

Return an iterator of the boxes of self that are in row jj but not identical with (ii, jj), or lie in the row jj - 1 (the row directly below jj; this might be the basement) and strictly to the right of (ii, jj).

EXAMPLES:

sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a = a.shape()
sage: list(a.boxes_same_and_lower_right(1,1))
[(2, 1), (3, 1), (6, 1), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0)]
sage: list(a.boxes_same_and_lower_right(1,2))
[(3, 2), (6, 2), (2, 1), (3, 1), (6, 1)]
sage: list(a.boxes_same_and_lower_right(3,3))
[(6, 2)]
sage: list(a.boxes_same_and_lower_right(2,3))
[(3, 3), (3, 2), (6, 2)]

>>> from sage.all import *
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> a = a.shape()
>>> list(a.boxes_same_and_lower_right(Integer(1),Integer(1)))
[(2, 1), (3, 1), (6, 1), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0)]
>>> list(a.boxes_same_and_lower_right(Integer(1),Integer(2)))
[(3, 2), (6, 2), (2, 1), (3, 1), (6, 1)]
>>> list(a.boxes_same_and_lower_right(Integer(3),Integer(3)))
[(6, 2)]
>>> list(a.boxes_same_and_lower_right(Integer(2),Integer(3)))
[(3, 3), (3, 2), (6, 2)]
flip()[source]#

Return the flip of self, where flip is defined as follows. Let r = max(self). Then self.flip()[i] = r - self[i].

EXAMPLES:

sage: a = LatticeDiagram([3,0,2])
sage: a.flip()
[0, 3, 1]

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(0),Integer(2)])
>>> a.flip()
[0, 3, 1]
l(i, j)[source]#

Return self[i] - j.

EXAMPLES:

sage: a = LatticeDiagram([3,1,2,4,3,0,4,2,3])
sage: a.l(5,2)
1

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(1),Integer(2),Integer(4),Integer(3),Integer(0),Integer(4),Integer(2),Integer(3)])
>>> a.l(Integer(5),Integer(2))
1
leg(i, j)[source]#

Return the leg of the box (i,j) in self.

EXAMPLES:

sage: a = LatticeDiagram([3,1,2,4,3,0,4,2,3])
sage: a.leg(5,2)
[(5, 3)]

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(1),Integer(2),Integer(4),Integer(3),Integer(0),Integer(4),Integer(2),Integer(3)])
>>> a.leg(Integer(5),Integer(2))
[(5, 3)]
size()[source]#

Return the number of boxes in self.

EXAMPLES:

sage: a = LatticeDiagram([3,1,2,4,3,0,4,2,3])
sage: a.size()
22

>>> from sage.all import *
>>> a = LatticeDiagram([Integer(3),Integer(1),Integer(2),Integer(4),Integer(3),Integer(0),Integer(4),Integer(2),Integer(3)])
>>> a.size()
22
class sage.combinat.sf.ns_macdonald.NonattackingBacktracker(shape, pi=None)[source]#

Bases: GenericBacktracker

EXAMPLES:

sage: from sage.combinat.sf.ns_macdonald import NonattackingBacktracker
sage: n = NonattackingBacktracker(LatticeDiagram([0,1,2]))
sage: n._ending_position
(3, 2)
sage: n._initial_state
(2, 1)

>>> from sage.all import *
>>> from sage.combinat.sf.ns_macdonald import NonattackingBacktracker
>>> n = NonattackingBacktracker(LatticeDiagram([Integer(0),Integer(1),Integer(2)]))
>>> n._ending_position
(3, 2)
>>> n._initial_state
(2, 1)
get_next_pos(ii, jj)[source]#

EXAMPLES:

sage: from sage.combinat.sf.ns_macdonald import NonattackingBacktracker
sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: n = NonattackingBacktracker(a.shape())
sage: n.get_next_pos(1, 1)
(2, 1)
sage: n.get_next_pos(6, 1)
(1, 2)
sage: n = NonattackingBacktracker(LatticeDiagram([2,2,2]))
sage: n.get_next_pos(3, 1)
(1, 2)

>>> from sage.all import *
>>> from sage.combinat.sf.ns_macdonald import NonattackingBacktracker
>>> a = AugmentedLatticeDiagramFilling([[Integer(1),Integer(6)],[Integer(2)],[Integer(3),Integer(4),Integer(2)],[],[],[Integer(5),Integer(5)]])
>>> n = NonattackingBacktracker(a.shape())
>>> n.get_next_pos(Integer(1), Integer(1))
(2, 1)
>>> n.get_next_pos(Integer(6), Integer(1))
(1, 2)
>>> n = NonattackingBacktracker(LatticeDiagram([Integer(2),Integer(2),Integer(2)]))
>>> n.get_next_pos(Integer(3), Integer(1))
(1, 2)
sage.combinat.sf.ns_macdonald.NonattackingFillings(shape, pi=None)[source]#

Returning the finite set of nonattacking fillings of a given shape.

EXAMPLES:

sage: NonattackingFillings([0,1,2])
Nonattacking fillings of [0, 1, 2]
sage: NonattackingFillings([0,1,2]).list()
[[[1], [2, 1], [3, 2, 1]],
 [[1], [2, 1], [3, 2, 2]],
 [[1], [2, 1], [3, 2, 3]],
 [[1], [2, 1], [3, 3, 1]],
 [[1], [2, 1], [3, 3, 2]],
 [[1], [2, 1], [3, 3, 3]],
 [[1], [2, 2], [3, 1, 1]],
 [[1], [2, 2], [3, 1, 2]],
 [[1], [2, 2], [3, 1, 3]],
 [[1], [2, 2], [3, 3, 1]],
 [[1], [2, 2], [3, 3, 2]],
 [[1], [2, 2], [3, 3, 3]]]

>>> from sage.all import *
>>> NonattackingFillings([Integer(0),Integer(1),Integer(2)])
Nonattacking fillings of [0, 1, 2]
>>> NonattackingFillings([Integer(0),Integer(1),Integer(2)]).list()
[[[1], [2, 1], [3, 2, 1]],
 [[1], [2, 1], [3, 2, 2]],
 [[1], [2, 1], [3, 2, 3]],
 [[1], [2, 1], [3, 3, 1]],
 [[1], [2, 1], [3, 3, 2]],
 [[1], [2, 1], [3, 3, 3]],
 [[1], [2, 2], [3, 1, 1]],
 [[1], [2, 2], [3, 1, 2]],
 [[1], [2, 2], [3, 1, 3]],
 [[1], [2, 2], [3, 3, 1]],
 [[1], [2, 2], [3, 3, 2]],
 [[1], [2, 2], [3, 3, 3]]]
class sage.combinat.sf.ns_macdonald.NonattackingFillings_shape(shape, pi=None)[source]#

Bases: Parent, UniqueRepresentation

EXAMPLES:

sage: n = NonattackingFillings([0,1,2])
sage: n == loads(dumps(n))
True

>>> from sage.all import *
>>> n = NonattackingFillings([Integer(0),Integer(1),Integer(2)])
>>> n == loads(dumps(n))
True
flip()[source]#

Return the nonattacking fillings of the flipped shape.

EXAMPLES:

sage: NonattackingFillings([0,1,2]).flip()
Nonattacking fillings of [2, 1, 0]

>>> from sage.all import *
>>> NonattackingFillings([Integer(0),Integer(1),Integer(2)]).flip()
Nonattacking fillings of [2, 1, 0]