Ribbon Tableaux#

class sage.combinat.ribbon_tableau.MultiSkewTableau(parent, *args, **kwds)[source]#

Bases: CombinatorialElement

A multi skew tableau which is a tuple of skew tableaux.

EXAMPLES:

sage: s = MultiSkewTableau([ [[None,1],[2,3]], [[1,2],[2]] ])
sage: s.size()
6
sage: s.weight()
[2, 3, 1]
sage: s.shape()
[[2, 2] / [1], [2, 1] / []]

>>> from sage.all import *
>>> s = MultiSkewTableau([ [[None,Integer(1)],[Integer(2),Integer(3)]], [[Integer(1),Integer(2)],[Integer(2)]] ])
>>> s.size()
6
>>> s.weight()
[2, 3, 1]
>>> s.shape()
[[2, 2] / [1], [2, 1] / []]
inversion_pairs()[source]#

Return a list of the inversion pairs of self.

EXAMPLES:

sage: s = MultiSkewTableau([ [[2,3],[5,5]], [[1,1],[3,3]], [[2],[6]] ])
sage: s.inversion_pairs()
[((0, (0, 0)), (1, (0, 0))),
 ((0, (1, 0)), (1, (0, 1))),
 ((0, (1, 1)), (1, (0, 0))),
 ((0, (1, 1)), (1, (1, 1))),
 ((0, (1, 1)), (2, (0, 0))),
 ((1, (0, 1)), (2, (0, 0))),
 ((1, (1, 1)), (2, (0, 0)))]

>>> from sage.all import *
>>> s = MultiSkewTableau([ [[Integer(2),Integer(3)],[Integer(5),Integer(5)]], [[Integer(1),Integer(1)],[Integer(3),Integer(3)]], [[Integer(2)],[Integer(6)]] ])
>>> s.inversion_pairs()
[((0, (0, 0)), (1, (0, 0))),
 ((0, (1, 0)), (1, (0, 1))),
 ((0, (1, 1)), (1, (0, 0))),
 ((0, (1, 1)), (1, (1, 1))),
 ((0, (1, 1)), (2, (0, 0))),
 ((1, (0, 1)), (2, (0, 0))),
 ((1, (1, 1)), (2, (0, 0)))]
inversions()[source]#

Return the number of inversion pairs of self.

EXAMPLES:

sage: t1 = SkewTableau([[1]])
sage: t2 = SkewTableau([[2]])
sage: MultiSkewTableau([t1,t1]).inversions()
0
sage: MultiSkewTableau([t1,t2]).inversions()
0
sage: MultiSkewTableau([t2,t2]).inversions()
0
sage: MultiSkewTableau([t2,t1]).inversions()
1
sage: s = MultiSkewTableau([ [[2,3],[5,5]], [[1,1],[3,3]], [[2],[6]] ])
sage: s.inversions()
7

>>> from sage.all import *
>>> t1 = SkewTableau([[Integer(1)]])
>>> t2 = SkewTableau([[Integer(2)]])
>>> MultiSkewTableau([t1,t1]).inversions()
0
>>> MultiSkewTableau([t1,t2]).inversions()
0
>>> MultiSkewTableau([t2,t2]).inversions()
0
>>> MultiSkewTableau([t2,t1]).inversions()
1
>>> s = MultiSkewTableau([ [[Integer(2),Integer(3)],[Integer(5),Integer(5)]], [[Integer(1),Integer(1)],[Integer(3),Integer(3)]], [[Integer(2)],[Integer(6)]] ])
>>> s.inversions()
7
shape()[source]#

Return the shape of self.

EXAMPLES:

sage: s = SemistandardSkewTableaux([[2,2],[1]]).list()
sage: a = MultiSkewTableau([s[0],s[1],s[2]])
sage: a.shape()
[[2, 2] / [1], [2, 2] / [1], [2, 2] / [1]]

>>> from sage.all import *
>>> s = SemistandardSkewTableaux([[Integer(2),Integer(2)],[Integer(1)]]).list()
>>> a = MultiSkewTableau([s[Integer(0)],s[Integer(1)],s[Integer(2)]])
>>> a.shape()
[[2, 2] / [1], [2, 2] / [1], [2, 2] / [1]]
size()[source]#

Return the size of self.

This is the sum of the sizes of the skew tableaux in self.

EXAMPLES:

sage: s = SemistandardSkewTableaux([[2,2],[1]]).list()
sage: a = MultiSkewTableau([s[0],s[1],s[2]])
sage: a.size()
9

>>> from sage.all import *
>>> s = SemistandardSkewTableaux([[Integer(2),Integer(2)],[Integer(1)]]).list()
>>> a = MultiSkewTableau([s[Integer(0)],s[Integer(1)],s[Integer(2)]])
>>> a.size()
9
weight()[source]#

Return the weight of self.

EXAMPLES:

sage: s = SemistandardSkewTableaux([[2,2],[1]]).list()
sage: a = MultiSkewTableau([s[0],s[1],s[2]])
sage: a.weight()
[5, 3, 1]

>>> from sage.all import *
>>> s = SemistandardSkewTableaux([[Integer(2),Integer(2)],[Integer(1)]]).list()
>>> a = MultiSkewTableau([s[Integer(0)],s[Integer(1)],s[Integer(2)]])
>>> a.weight()
[5, 3, 1]
class sage.combinat.ribbon_tableau.MultiSkewTableaux(category=None)[source]#

Bases: UniqueRepresentation, Parent

Multiskew tableaux.

Element[source]#

alias of MultiSkewTableau

class sage.combinat.ribbon_tableau.RibbonTableau(parent, st)[source]#

Bases: SkewTableau

A ribbon tableau.

A ribbon is a connected skew shape which does not contain any \(2 \times 2\) boxes. A ribbon tableau is a skew tableau whose shape is partitioned into ribbons, each of which is filled with identical entries.

EXAMPLES:

sage: rt = RibbonTableau([[None, 1],[2,3]]); rt
[[None, 1], [2, 3]]
sage: rt.inner_shape()
[1]
sage: rt.outer_shape()
[2, 2]

sage: rt = RibbonTableau([[None, None, 0, 0, 0], [None, 0, 0, 2], [1, 0, 1]]); rt.pp()
  .  .  0  0  0
  .  0  0  2
  1  0  1

>>> from sage.all import *
>>> rt = RibbonTableau([[None, Integer(1)],[Integer(2),Integer(3)]]); rt
[[None, 1], [2, 3]]
>>> rt.inner_shape()
[1]
>>> rt.outer_shape()
[2, 2]

>>> rt = RibbonTableau([[None, None, Integer(0), Integer(0), Integer(0)], [None, Integer(0), Integer(0), Integer(2)], [Integer(1), Integer(0), Integer(1)]]); rt.pp()
  .  .  0  0  0
  .  0  0  2
  1  0  1

In the previous example, each ribbon is uniquely determined by a non-zero entry. The 0 entries are used to fill in the rest of the skew shape.

Note

Sanity checks are not performed; lists can contain any object.

sage: RibbonTableau(expr=[[1,1],[[5],[3,4],[1,2]]])
[[None, 1, 2], [None, 3, 4], [5]]

>>> from sage.all import *
>>> RibbonTableau(expr=[[Integer(1),Integer(1)],[[Integer(5)],[Integer(3),Integer(4)],[Integer(1),Integer(2)]]])
[[None, 1, 2], [None, 3, 4], [5]]
length()[source]#

Return the length of the ribbons into a ribbon tableau.

EXAMPLES:

sage: RibbonTableau([[None, 1],[2,3]]).length()
1
sage: RibbonTableau([[1,0],[2,0]]).length()
2

>>> from sage.all import *
>>> RibbonTableau([[None, Integer(1)],[Integer(2),Integer(3)]]).length()
1
>>> RibbonTableau([[Integer(1),Integer(0)],[Integer(2),Integer(0)]]).length()
2
to_word()[source]#

Return a word obtained from a row reading of self.

Warning

Unlike the to_word method on skew tableaux (which are a superclass of this), this method does not filter out None entries.

EXAMPLES:

sage: R = RibbonTableau([[0, 0, 3, 0], [1, 1, 0], [2, 0, 4]])
sage: R.to_word()
word: 2041100030

>>> from sage.all import *
>>> R = RibbonTableau([[Integer(0), Integer(0), Integer(3), Integer(0)], [Integer(1), Integer(1), Integer(0)], [Integer(2), Integer(0), Integer(4)]])
>>> R.to_word()
word: 2041100030
class sage.combinat.ribbon_tableau.RibbonTableau_class(parent, st)[source]#

Bases: RibbonTableau

This exists solely for unpickling RibbonTableau_class objects.

class sage.combinat.ribbon_tableau.RibbonTableaux[source]#

Bases: UniqueRepresentation, Parent

Ribbon tableaux.

A ribbon tableau is a skew tableau whose skew shape shape is tiled by ribbons of length length. The weight weight is calculated from the labels on the ribbons.

Note

Here we impose the condition that the ribbon tableaux are semistandard.

INPUT(Optional):

  • shape – skew shape as a list of lists or an object of type SkewPartition

  • length – integer, shape is partitioned into ribbons of length length

  • weight – list of integers, computed from the values of non-zero entries labeling the ribbons

EXAMPLES:

sage: RibbonTableaux([[2,1],[]], [1,1,1], 1)
Ribbon tableaux of shape [2, 1] / [] and weight [1, 1, 1] with 1-ribbons

sage: R = RibbonTableaux([[5,4,3],[2,1]], [2,1], 3)
sage: for i in R: i.pp(); print("\n")
  .  .  0  0  0
  .  0  0  2
  1  0  1

  .  .  1  0  0
  .  0  0  0
  1  0  2

  .  .  0  0  0
  .  1  0  1
  2  0  0

>>> from sage.all import *
>>> RibbonTableaux([[Integer(2),Integer(1)],[]], [Integer(1),Integer(1),Integer(1)], Integer(1))
Ribbon tableaux of shape [2, 1] / [] and weight [1, 1, 1] with 1-ribbons

>>> R = RibbonTableaux([[Integer(5),Integer(4),Integer(3)],[Integer(2),Integer(1)]], [Integer(2),Integer(1)], Integer(3))
>>> for i in R: i.pp(); print("\n")
  .  .  0  0  0
  .  0  0  2
  1  0  1
<BLANKLINE>
  .  .  1  0  0
  .  0  0  0
  1  0  2
<BLANKLINE>
  .  .  0  0  0
  .  1  0  1
  2  0  0
<BLANKLINE>

REFERENCES:

[vanLeeuwen91]

Marc. A. A. van Leeuwen, Edge sequences, ribbon tableaux, and an action of affine permutations. Europe J. Combinatorics. 20 (1999). http://wwwmathlabo.univ-poitiers.fr/~maavl/pdf/edgeseqs.pdf

Element[source]#

alias of RibbonTableau

from_expr(l)[source]#

Return a RibbonTableau from a MuPAD-Combinat expr for a skew tableau. The first list in expr is the inner shape of the skew tableau. The second list are the entries in the rows of the skew tableau from bottom to top.

Provided primarily for compatibility with MuPAD-Combinat.

EXAMPLES:

sage: RibbonTableaux().from_expr([[1,1],[[5],[3,4],[1,2]]])
[[None, 1, 2], [None, 3, 4], [5]]

>>> from sage.all import *
>>> RibbonTableaux().from_expr([[Integer(1),Integer(1)],[[Integer(5)],[Integer(3),Integer(4)],[Integer(1),Integer(2)]]])
[[None, 1, 2], [None, 3, 4], [5]]
options = Current options for Tableaux   - ascii_art:  repr   - convention: English   - display:    list   - latex:      diagram[source]#
class sage.combinat.ribbon_tableau.RibbonTableaux_shape_weight_length(shape, weight, length)[source]#

Bases: RibbonTableaux

Ribbon tableaux of a given shape, weight, and length.

cardinality()[source]#

Return the cardinality of self.

EXAMPLES:

sage: RibbonTableaux([[2,1],[]],[1,1,1],1).cardinality()
2
sage: RibbonTableaux([[2,2],[]],[1,1],2).cardinality()
2
sage: RibbonTableaux([[4,3,3],[]],[2,1,1,1],2).cardinality()
5

>>> from sage.all import *
>>> RibbonTableaux([[Integer(2),Integer(1)],[]],[Integer(1),Integer(1),Integer(1)],Integer(1)).cardinality()
2
>>> RibbonTableaux([[Integer(2),Integer(2)],[]],[Integer(1),Integer(1)],Integer(2)).cardinality()
2
>>> RibbonTableaux([[Integer(4),Integer(3),Integer(3)],[]],[Integer(2),Integer(1),Integer(1),Integer(1)],Integer(2)).cardinality()
5
class sage.combinat.ribbon_tableau.SemistandardMultiSkewTableaux(shape, weight)[source]#

Bases: MultiSkewTableaux

Semistandard multi skew tableaux.

A multi skew tableau is a \(k\)-tuple of skew tableaux of given shape with a specified total weight.

EXAMPLES:

sage: S = SemistandardMultiSkewTableaux([ [[2,1],[]], [[2,2],[1]] ], [2,2,2]); S
Semistandard multi skew tableaux of shape [[2, 1] / [], [2, 2] / [1]] and weight [2, 2, 2]
sage: S.list()
[[[[1, 1], [2]], [[None, 2], [3, 3]]],
 [[[1, 2], [2]], [[None, 1], [3, 3]]],
 [[[1, 3], [2]], [[None, 2], [1, 3]]],
 [[[1, 3], [2]], [[None, 1], [2, 3]]],
 [[[1, 1], [3]], [[None, 2], [2, 3]]],
 [[[1, 2], [3]], [[None, 2], [1, 3]]],
 [[[1, 2], [3]], [[None, 1], [2, 3]]],
 [[[2, 2], [3]], [[None, 1], [1, 3]]],
 [[[1, 3], [3]], [[None, 1], [2, 2]]],
 [[[2, 3], [3]], [[None, 1], [1, 2]]]]

>>> from sage.all import *
>>> S = SemistandardMultiSkewTableaux([ [[Integer(2),Integer(1)],[]], [[Integer(2),Integer(2)],[Integer(1)]] ], [Integer(2),Integer(2),Integer(2)]); S
Semistandard multi skew tableaux of shape [[2, 1] / [], [2, 2] / [1]] and weight [2, 2, 2]
>>> S.list()
[[[[1, 1], [2]], [[None, 2], [3, 3]]],
 [[[1, 2], [2]], [[None, 1], [3, 3]]],
 [[[1, 3], [2]], [[None, 2], [1, 3]]],
 [[[1, 3], [2]], [[None, 1], [2, 3]]],
 [[[1, 1], [3]], [[None, 2], [2, 3]]],
 [[[1, 2], [3]], [[None, 2], [1, 3]]],
 [[[1, 2], [3]], [[None, 1], [2, 3]]],
 [[[2, 2], [3]], [[None, 1], [1, 3]]],
 [[[1, 3], [3]], [[None, 1], [2, 2]]],
 [[[2, 3], [3]], [[None, 1], [1, 2]]]]
sage.combinat.ribbon_tableau.cospin_polynomial(part, weight, length)[source]#

Return the cospin polynomial associated to part, weight, and length.

EXAMPLES:

sage: from sage.combinat.ribbon_tableau import cospin_polynomial
sage: cospin_polynomial([6,6,6],[4,2],3)
t^4 + t^3 + 2*t^2 + t + 1
sage: cospin_polynomial([3,3,3,2,1], [3,1], 3)
1
sage: cospin_polynomial([3,3,3,2,1], [2,2], 3)
t + 1
sage: cospin_polynomial([3,3,3,2,1], [2,1,1], 3)
t^2 + 2*t + 2
sage: cospin_polynomial([3,3,3,2,1], [1,1,1,1], 3)
t^3 + 3*t^2 + 5*t + 3
sage: cospin_polynomial([5,4,3,2,1,1,1], [2,2,1], 3)
2*t^2 + 6*t + 2
sage: cospin_polynomial([[6]*6, [3,3]], [4,4,2], 3)
3*t^4 + 6*t^3 + 9*t^2 + 5*t + 3

>>> from sage.all import *
>>> from sage.combinat.ribbon_tableau import cospin_polynomial
>>> cospin_polynomial([Integer(6),Integer(6),Integer(6)],[Integer(4),Integer(2)],Integer(3))
t^4 + t^3 + 2*t^2 + t + 1
>>> cospin_polynomial([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(3),Integer(1)], Integer(3))
1
>>> cospin_polynomial([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(2),Integer(2)], Integer(3))
t + 1
>>> cospin_polynomial([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(2),Integer(1),Integer(1)], Integer(3))
t^2 + 2*t + 2
>>> cospin_polynomial([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(1),Integer(1),Integer(1),Integer(1)], Integer(3))
t^3 + 3*t^2 + 5*t + 3
>>> cospin_polynomial([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1),Integer(1),Integer(1)], [Integer(2),Integer(2),Integer(1)], Integer(3))
2*t^2 + 6*t + 2
>>> cospin_polynomial([[Integer(6)]*Integer(6), [Integer(3),Integer(3)]], [Integer(4),Integer(4),Integer(2)], Integer(3))
3*t^4 + 6*t^3 + 9*t^2 + 5*t + 3
sage.combinat.ribbon_tableau.count_rec(nexts, current, part, weight, length)[source]#

INPUT:

  • nexts, current, part – skew partitions

  • weight – non-negative integer list

  • length – integer

sage.combinat.ribbon_tableau.graph_implementation_rec(skp, weight, length, function)[source]#
sage.combinat.ribbon_tableau.insertion_tableau(skp, perm, evaluation, tableau, length)[source]#

INPUT:

  • skp – skew partitions

  • perm, evaluation – non-negative integers

  • tableau – skew tableau

  • length – integer

sage.combinat.ribbon_tableau.list_rec(nexts, current, part, weight, length)[source]#

INPUT:

  • nexts, current, part – skew partitions

  • weight – non-negative integer list

  • length – integer

sage.combinat.ribbon_tableau.spin_polynomial(part, weight, length)[source]#

Returns the spin polynomial associated to part, weight, and length.

EXAMPLES:

sage: # needs sage.symbolic
sage: from sage.combinat.ribbon_tableau import spin_polynomial
sage: spin_polynomial([6,6,6],[4,2],3)
t^6 + t^5 + 2*t^4 + t^3 + t^2
sage: spin_polynomial([6,6,6],[4,1,1],3)
t^6 + 2*t^5 + 3*t^4 + 2*t^3 + t^2
sage: spin_polynomial([3,3,3,2,1], [2,2], 3)
t^(7/2) + t^(5/2)
sage: spin_polynomial([3,3,3,2,1], [2,1,1], 3)
2*t^(7/2) + 2*t^(5/2) + t^(3/2)
sage: spin_polynomial([3,3,3,2,1], [1,1,1,1], 3)
3*t^(7/2) + 5*t^(5/2) + 3*t^(3/2) + sqrt(t)
sage: spin_polynomial([5,4,3,2,1,1,1], [2,2,1], 3)
2*t^(9/2) + 6*t^(7/2) + 2*t^(5/2)
sage: spin_polynomial([[6]*6, [3,3]], [4,4,2], 3)
3*t^9 + 5*t^8 + 9*t^7 + 6*t^6 + 3*t^5

>>> from sage.all import *
>>> # needs sage.symbolic
>>> from sage.combinat.ribbon_tableau import spin_polynomial
>>> spin_polynomial([Integer(6),Integer(6),Integer(6)],[Integer(4),Integer(2)],Integer(3))
t^6 + t^5 + 2*t^4 + t^3 + t^2
>>> spin_polynomial([Integer(6),Integer(6),Integer(6)],[Integer(4),Integer(1),Integer(1)],Integer(3))
t^6 + 2*t^5 + 3*t^4 + 2*t^3 + t^2
>>> spin_polynomial([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(2),Integer(2)], Integer(3))
t^(7/2) + t^(5/2)
>>> spin_polynomial([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(2),Integer(1),Integer(1)], Integer(3))
2*t^(7/2) + 2*t^(5/2) + t^(3/2)
>>> spin_polynomial([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(1),Integer(1),Integer(1),Integer(1)], Integer(3))
3*t^(7/2) + 5*t^(5/2) + 3*t^(3/2) + sqrt(t)
>>> spin_polynomial([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1),Integer(1),Integer(1)], [Integer(2),Integer(2),Integer(1)], Integer(3))
2*t^(9/2) + 6*t^(7/2) + 2*t^(5/2)
>>> spin_polynomial([[Integer(6)]*Integer(6), [Integer(3),Integer(3)]], [Integer(4),Integer(4),Integer(2)], Integer(3))
3*t^9 + 5*t^8 + 9*t^7 + 6*t^6 + 3*t^5
sage.combinat.ribbon_tableau.spin_polynomial_square(part, weight, length)[source]#

Returns the spin polynomial associated with part, weight, and length, with the substitution \(t \to t^2\) made.

EXAMPLES:

sage: from sage.combinat.ribbon_tableau import spin_polynomial_square
sage: spin_polynomial_square([6,6,6],[4,2],3)
t^12 + t^10 + 2*t^8 + t^6 + t^4
sage: spin_polynomial_square([6,6,6],[4,1,1],3)
t^12 + 2*t^10 + 3*t^8 + 2*t^6 + t^4
sage: spin_polynomial_square([3,3,3,2,1], [2,2], 3)
t^7 + t^5
sage: spin_polynomial_square([3,3,3,2,1], [2,1,1], 3)
2*t^7 + 2*t^5 + t^3
sage: spin_polynomial_square([3,3,3,2,1], [1,1,1,1], 3)
3*t^7 + 5*t^5 + 3*t^3 + t
sage: spin_polynomial_square([5,4,3,2,1,1,1], [2,2,1], 3)
2*t^9 + 6*t^7 + 2*t^5
sage: spin_polynomial_square([[6]*6, [3,3]], [4,4,2], 3)
3*t^18 + 5*t^16 + 9*t^14 + 6*t^12 + 3*t^10

>>> from sage.all import *
>>> from sage.combinat.ribbon_tableau import spin_polynomial_square
>>> spin_polynomial_square([Integer(6),Integer(6),Integer(6)],[Integer(4),Integer(2)],Integer(3))
t^12 + t^10 + 2*t^8 + t^6 + t^4
>>> spin_polynomial_square([Integer(6),Integer(6),Integer(6)],[Integer(4),Integer(1),Integer(1)],Integer(3))
t^12 + 2*t^10 + 3*t^8 + 2*t^6 + t^4
>>> spin_polynomial_square([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(2),Integer(2)], Integer(3))
t^7 + t^5
>>> spin_polynomial_square([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(2),Integer(1),Integer(1)], Integer(3))
2*t^7 + 2*t^5 + t^3
>>> spin_polynomial_square([Integer(3),Integer(3),Integer(3),Integer(2),Integer(1)], [Integer(1),Integer(1),Integer(1),Integer(1)], Integer(3))
3*t^7 + 5*t^5 + 3*t^3 + t
>>> spin_polynomial_square([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1),Integer(1),Integer(1)], [Integer(2),Integer(2),Integer(1)], Integer(3))
2*t^9 + 6*t^7 + 2*t^5
>>> spin_polynomial_square([[Integer(6)]*Integer(6), [Integer(3),Integer(3)]], [Integer(4),Integer(4),Integer(2)], Integer(3))
3*t^18 + 5*t^16 + 9*t^14 + 6*t^12 + 3*t^10
sage.combinat.ribbon_tableau.spin_rec(t, nexts, current, part, weight, length)[source]#

Routine used for constructing the spin polynomial.

INPUT:

  • weight – list of non-negative integers

  • length – the length of the ribbons we’re tiling with

  • t – the variable

EXAMPLES:

sage: from sage.combinat.ribbon_tableau import spin_rec
sage: sp = SkewPartition
sage: t = ZZ['t'].gen()
sage: spin_rec(t, [], [[[], [3, 3]]], sp([[2, 2, 2], []]), [2], 3)
[t^4]
sage: spin_rec(t, [[0], [t^4]], [[[2, 1, 1, 1, 1], [0, 3]], [[2, 2, 2], [3, 0]]], sp([[2, 2, 2, 2, 1], []]), [2, 1], 3)
[t^5]
sage: spin_rec(t, [], [[[], [3, 3, 0]]], sp([[3, 3], []]), [2], 3)
[t^2]
sage: spin_rec(t, [[t^4], [t^3], [t^2]], [[[2, 2, 2], [0, 0, 3]], [[3, 2, 1], [0, 3, 0]], [[3, 3], [3, 0, 0]]], sp([[3, 3, 3], []]), [2, 1], 3)
[t^6 + t^4 + t^2]
sage: spin_rec(t, [[t^5], [t^4], [t^6 + t^4 + t^2]], [[[2, 2, 2, 2, 1], [0, 0, 3]], [[3, 3, 1, 1, 1], [0, 3, 0]], [[3, 3, 3], [3, 0, 0]]], sp([[3, 3, 3, 2, 1], []]), [2, 1, 1], 3)
[2*t^7 + 2*t^5 + t^3]

>>> from sage.all import *
>>> from sage.combinat.ribbon_tableau import spin_rec
>>> sp = SkewPartition
>>> t = ZZ['t'].gen()
>>> spin_rec(t, [], [[[], [Integer(3), Integer(3)]]], sp([[Integer(2), Integer(2), Integer(2)], []]), [Integer(2)], Integer(3))
[t^4]
>>> spin_rec(t, [[Integer(0)], [t**Integer(4)]], [[[Integer(2), Integer(1), Integer(1), Integer(1), Integer(1)], [Integer(0), Integer(3)]], [[Integer(2), Integer(2), Integer(2)], [Integer(3), Integer(0)]]], sp([[Integer(2), Integer(2), Integer(2), Integer(2), Integer(1)], []]), [Integer(2), Integer(1)], Integer(3))
[t^5]
>>> spin_rec(t, [], [[[], [Integer(3), Integer(3), Integer(0)]]], sp([[Integer(3), Integer(3)], []]), [Integer(2)], Integer(3))
[t^2]
>>> spin_rec(t, [[t**Integer(4)], [t**Integer(3)], [t**Integer(2)]], [[[Integer(2), Integer(2), Integer(2)], [Integer(0), Integer(0), Integer(3)]], [[Integer(3), Integer(2), Integer(1)], [Integer(0), Integer(3), Integer(0)]], [[Integer(3), Integer(3)], [Integer(3), Integer(0), Integer(0)]]], sp([[Integer(3), Integer(3), Integer(3)], []]), [Integer(2), Integer(1)], Integer(3))
[t^6 + t^4 + t^2]
>>> spin_rec(t, [[t**Integer(5)], [t**Integer(4)], [t**Integer(6) + t**Integer(4) + t**Integer(2)]], [[[Integer(2), Integer(2), Integer(2), Integer(2), Integer(1)], [Integer(0), Integer(0), Integer(3)]], [[Integer(3), Integer(3), Integer(1), Integer(1), Integer(1)], [Integer(0), Integer(3), Integer(0)]], [[Integer(3), Integer(3), Integer(3)], [Integer(3), Integer(0), Integer(0)]]], sp([[Integer(3), Integer(3), Integer(3), Integer(2), Integer(1)], []]), [Integer(2), Integer(1), Integer(1)], Integer(3))
[2*t^7 + 2*t^5 + t^3]