Shifted primed tableaux¶
AUTHORS:
Kirill Paramonov (2017-08-18): initial implementation
Chaman Agrawal (2019-08-12): add parameter to allow primed diagonal entry
- class sage.combinat.shifted_primed_tableau.CrystalElementShiftedPrimedTableau(parent, T, skew=None, check=True, preprocessed=False)¶
Bases:
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableauClass for elements of
crystals.ShiftedPrimedTableau.- e(ind)¶
Compute the action of the crystal operator \(e_i\) on a shifted primed tableau using cases from the papers [HPS2017] and [AO2018].
INPUT:
ind– an element in the index set of the crystal
OUTPUT:
Primed tableau or
None.EXAMPLES:
sage: SPT = ShiftedPrimedTableaux([5,4,2]) sage: t = SPT([[1,1,1,'2p','3p'], [2,'3p',3,3],[3,4]]) sage: t.pp() 1 1 1 2' 3' 2 3' 3 3 3 4 sage: s = t.e(2) sage: s.pp() 1 1 1 2' 3' 2 2 3 3 3 4 sage: t == s.f(2) True sage: SPT = ShiftedPrimedTableaux([2,1]) sage: t = SPT([[2,'3p'],[3]]) sage: t.e(-1).pp() 1 3' 3 sage: t.e(1).pp() 1 3' 3 sage: t.e(2).pp() 2 2 3 sage: r = SPT([[2, 2],[3]]) sage: r.e(-1).pp() 1 2 3 sage: r.e(1).pp() 1 2 3 sage: r.e(2) is None True sage: r = SPT([[1,'3p'],[3]]) sage: r.e(-1) is None True sage: r.e(1) is None True sage: r.e(2).pp() 1 2' 3 sage: r = SPT([[1,'2p'],[3]]) sage: r.e(-1).pp() 1 1 3 sage: r.e(1) is None True sage: r.e(2).pp() 1 2' 2 sage: t = SPT([[2,'3p'],[3]]) sage: t.e(-1).e(2).e(2).e(-1) == t.e(2).e(1).e(1).e(2) True sage: t.e(-1).e(2).e(2).e(-1).pp() 1 1 2 sage: all(t.e(-1).e(2).e(2).e(-1).e(i) is None for i in {-1, 1, 2}) True sage: SPT = ShiftedPrimedTableaux([4]) sage: t = SPT([[2,2,2,2]]) sage: t.e(-1).pp() 1 2 2 2 sage: t.e(1).pp() 1 2 2 2 sage: t.e(-1).e(-1) is None True sage: t.e(1).e(1).pp() 1 1 2 2
- f(ind)¶
Compute the action of the crystal operator \(f_i\) on a shifted primed tableau using cases from the papers [HPS2017] and [AO2018].
INPUT:
ind– element in the index set of the crystal
OUTPUT:
Primed tableau or
None.EXAMPLES:
sage: SPT = ShiftedPrimedTableaux([5,4,2]) sage: t = SPT([[1,1,1,1,'3p'],[2,2,2,'3p'],[3,3]]) sage: t.pp() 1 1 1 1 3' 2 2 2 3' 3 3 sage: s = t.f(2) sage: s is None True sage: t = SPT([[1,1,1,'2p','3p'],[2,2,3,3],[3,4]]) sage: t.pp() 1 1 1 2' 3' 2 2 3 3 3 4 sage: s = t.f(2) sage: s.pp() 1 1 1 2' 3' 2 3' 3 3 3 4 sage: SPT = ShiftedPrimedTableaux([2,1]) sage: t = SPT([[1,1],[2]]) sage: t.f(-1).pp() 1 2' 2 sage: t.f(1).pp() 1 2' 2 sage: t.f(2).pp() 1 1 3 sage: r = SPT([[1,'2p'],[2]]) sage: r.f(-1) is None True sage: r.f(1) is None True sage: r.f(2).pp() 1 2' 3 sage: r = SPT([[1,1],[3]]) sage: r.f(-1).pp() 1 2' 3 sage: r.f(1).pp() 1 2 3 sage: r.f(2) is None True sage: r = SPT([[1,2],[3]]) sage: r.f(-1).pp() 2 2 3 sage: r.f(1).pp() 2 2 3 sage: r.f(2) is None True sage: t = SPT([[1,1],[2]]) sage: t.f(-1).f(2).f(2).f(-1) == t.f(2).f(1).f(-1).f(2) True sage: t.f(-1).f(2).f(2).f(-1).pp() 2 3' 3 sage: all(t.f(-1).f(2).f(2).f(-1).f(i) is None for i in {-1, 1, 2}) True sage: SPT = ShiftedPrimedTableaux([4]) sage: t = SPT([[1,1,1,1]]) sage: t.f(-1).pp() 1 1 1 2' sage: t.f(1).pp() 1 1 1 2 sage: t.f(-1).f(-1) is None True sage: t.f(1).f(-1).pp() 1 1 2' 2 sage: t.f(1).f(1).pp() 1 1 2 2 sage: t.f(1).f(1).f(-1).pp() 1 2' 2 2 sage: t.f(1).f(1).f(1).pp() 1 2 2 2 sage: t.f(1).f(1).f(1).f(-1).pp() 2 2 2 2 sage: t.f(1).f(1).f(1).f(1).pp() 2 2 2 2 sage: t.f(1).f(1).f(1).f(1).f(-1) is None True
- is_highest_weight(index_set=None)¶
Return whether
selfis a highest weight element of the crystal.An element is highest weight if it vanishes under all crystal operators \(e_i\).
EXAMPLES:
sage: SPT = ShiftedPrimedTableaux([5,4,2]) sage: t = SPT([(1, 1, 1, 1, 1), (2, 2, 2, "3p"), (3, 3)]) sage: t.is_highest_weight() True sage: SPT = ShiftedPrimedTableaux([5,4]) sage: s = SPT([(1, 1, 1, 1, 1), (2, 2, "3p", 3)]) sage: s.is_highest_weight(index_set=[1]) True
- reading_word()¶
Return the reading word of
self.The reading word of a shifted primed tableau is constructed as follows:
List all primed entries in the tableau, column by column, in decreasing order within each column, moving from the rightmost column to the left, and with all the primes removed (i.e. all entries are increased by half a unit).
Then list all unprimed entries, row by row, in increasing order within each row, moving from the bottommost row to the top.
EXAMPLES:
sage: SPT = ShiftedPrimedTableaux([4,2]) sage: t = SPT([[1,'2p',2,2],[2,'3p']]) sage: t.reading_word() [3, 2, 2, 1, 2, 2]
- weight()¶
Return the weight of
self.The weight of a shifted primed tableau is defined to be the vector with \(i\)-th component equal to the number of entries \(i\) and \(i'\) in the tableau.
EXAMPLES:
sage: t = ShiftedPrimedTableau([[1,'2p',2,2],[2,'3p']]) sage: t.weight() (1, 4, 1)
- class sage.combinat.shifted_primed_tableau.PrimedEntry(entry=None, double=None)¶
Bases:
sage.structure.sage_object.SageObjectThe class of entries in shifted primed tableaux.
An entry in a shifted primed tableau is an element in the alphabet \(\{1' < 1 < 2' < 2 < \cdots < n' < n\}\). The difference between two elements \(i\) and \(i-1\) counts as a whole unit, whereas the difference between \(i\) and \(i'\) counts as half a unit. Internally, we represent an unprimed element \(x\) as \(2x\) and the primed elements as the corresponding odd integer that respects the total order.
INPUT:
entry– a half integer or a string of an integer possibly ending inpor'double– the doubled value
- decrease_half()¶
Decrease
selfby half a unit.
- decrease_one()¶
Decrease
selfby one unit.
- increase_half()¶
Increase
selfby half a unit.
- increase_one()¶
Increase
selfby one unit.
- integer()¶
Return the corresponding integer \(i\) for primed entries of the form \(i\) or \(i'\).
- is_primed()¶
Checks if
selfis a primed element.
- is_unprimed()¶
Checks if
selfis an unprimed element.
- primed()¶
Prime
selfif it is an unprimed element.
- unprimed()¶
Unprime
selfif it is a primed element.
- class sage.combinat.shifted_primed_tableau.ShiftedPrimedTableau(parent, T, skew=None, check=True, preprocessed=False)¶
Bases:
sage.structure.list_clone.ClonableArrayA shifted primed tableau.
A primed tableau is a tableau of shifted shape in the alphabet \(X' = \{1' < 1 < 2' < 2 < \cdots < n' < n\}\) such that
the entries are weakly increasing along rows and columns;
a row cannot have two repeated primed elements, and a column cannot have two repeated non-primed elements;
Skew shape of the shifted primed tableaux is specified either with an optional argument
skewor withNoneentries.Primed entries in the main diagonal can be allowed with the optional boolean parameter
primed_diagonal``(default: ``False).EXAMPLES:
sage: T = ShiftedPrimedTableaux([4,2]) sage: T([[1,"2'","3'",3],[2,"3'"]])[1] (2, 3') sage: t = ShiftedPrimedTableau([[1,"2p",2.5,3],[2,2.5]]) sage: t[1] (2, 3') sage: ShiftedPrimedTableau([["2p",2,3],["2p","3p"],[2]], skew=[2,1]) [(None, None, 2', 2, 3), (None, 2', 3'), (2,)] sage: ShiftedPrimedTableau([[None,None,"2p"],[None,"2p"]]) [(None, None, 2'), (None, 2')] sage: T = ShiftedPrimedTableaux([4,2], primed_diagonal=True) sage: T([[1,"2'","3'",3],["2'","3'"]])[1] # With primed diagonal entry (2', 3')
- check()¶
Check that
selfis a valid primed tableau.EXAMPLES:
sage: T = ShiftedPrimedTableaux([4,2]) sage: t = T([[1,'2p',2,2],[2,'3p']]) sage: t.check() sage: s = ShiftedPrimedTableau([["2p",2,3],["2p"],[2]],skew=[2,1]) sage: s.check() sage: t = T([['1p','2p',2,2],[2,'3p']]) Traceback (most recent call last): ... ValueError: [['1p', '2p', 2, 2], [2, '3p']] is not an element of Shifted Primed Tableaux of shape [4, 2] sage: T = ShiftedPrimedTableaux([4,2], primed_diagonal=True) sage: t = T([['1p','2p',2,2],[2,'3p']]) # primed_diagonal allowed sage: t.check() sage: t = T([['1p','1p',2,2],[2,'3p']]) Traceback (most recent call last): ... ValueError: [['1p', '1p', 2, 2], [2, '3p']] is not an element of Shifted Primed Tableaux of shape [4, 2] and maximum entry 6
- is_standard()¶
Return
Trueif the entries ofselfare in bijection with positive primed integers \(1', 1, 2', \ldots, n\).EXAMPLES:
sage: ShiftedPrimedTableau([["1'", 1, "2'"], [2, "3'"]], ....: primed_diagonal=True).is_standard() True sage: ShiftedPrimedTableau([["1'", 1, 2], ["2'", "3'"]], ....: primed_diagonal=True).is_standard() True sage: ShiftedPrimedTableau([["1'", 1, 1], ["2'", 2]], ....: primed_diagonal=True).is_standard() False sage: ShiftedPrimedTableau([[1, "2'"], [2]]).is_standard() False sage: s = ShiftedPrimedTableau([[None, None,"1p","2p",2],[None,"1"]]) sage: s.is_standard() True
- max_entry()¶
Return the minimum unprimed letter \(x > y\) for all \(y\) in
self.EXAMPLES:
sage: Tab = ShiftedPrimedTableau([(1,1,'2p','3p'),(2,2)]) sage: Tab.max_entry() 3
- pp()¶
Pretty print
self.EXAMPLES:
sage: t = ShiftedPrimedTableau([[1,'2p',2,2],[2,'3p']]) sage: t.pp() 1 2' 2 2 2 3' sage: t = ShiftedPrimedTableau([[10,'11p',11,11],[11,'12']]) sage: t.pp() 10 11' 11 11 11 12 sage: s = ShiftedPrimedTableau([['2p',2,3],['2p']],skew=[2,1]) sage: s.pp() . . 2' 2 3 . 2'
- restrict(n)¶
Return the restriction of the shifted tableau to all the numbers less than or equal to
n.Note
If only the outer shape of the restriction, rather than the whole restriction, is needed, then the faster method
restriction_outer_shape()is preferred. Similarly if only the skew shape is needed, userestriction_shape().EXAMPLES:
sage: t = ShiftedPrimedTableau([[1,'2p',2,2],[2,'3p']]) sage: t.restrict(2).pp() 1 2' 2 2 2 sage: t.restrict("2p").pp() 1 2' sage: s = ShiftedPrimedTableau([["2p",2,3],["2p"]], skew=[2,1]) sage: s.restrict(2).pp() . . 2' 2 . 2' sage: s.restrict(1.5).pp() . . 2' . 2'
- restriction_outer_shape(n)¶
Return the outer shape of the restriction of the shifted tableau
selfto \(n\).If \(T\) is a (skew) shifted tableau and \(n\) is a half-integer, then the restriction of \(T\) to \(n\) is defined as the (skew) shifted tableau obtained by removing all cells filled with entries greater than \(n\) from \(T\).
This method computes merely the outer shape of the restriction. For the restriction itself, use
restrict().EXAMPLES:
sage: s = ShiftedPrimedTableau([["2p",2,3],["2p"]], skew=[2,1]) sage: s.pp() . . 2' 2 3 . 2' sage: s.restriction_outer_shape(2) [4, 2] sage: s.restriction_outer_shape("2p") [3, 2]
- restriction_shape(n)¶
Return the skew shape of the restriction of the skew tableau
selfton.If \(T\) is a shifted tableau and \(n\) is a half-integer, then the restriction of \(T\) to \(n\) is defined as the (skew) shifted tableau obtained by removing all cells filled with entries greater than \(n\) from \(T\).
This method computes merely the skew shape of the restriction. For the restriction itself, use
restrict().EXAMPLES:
sage: s = ShiftedPrimedTableau([["2p",2,3],["2p"]], skew=[2,1]) sage: s.pp() . . 2' 2 3 . 2' sage: s.restriction_shape(2) [4, 2] / [2, 1]
- shape()¶
Return the shape of the underlying partition of
self.EXAMPLES:
sage: t = ShiftedPrimedTableau([[1,'2p',2,2],[2,'3p']]) sage: t.shape() [4, 2] sage: s = ShiftedPrimedTableau([["2p",2,3],["2p"]],skew=[2,1]) sage: s.shape() [5, 2] / [2, 1]
- to_chain()¶
Return the chain of partitions corresponding to the (skew) shifted tableau
self, interlaced by one of the colours1is the added cell is on the diagonal,2if an ordinary entry is added and3if a primed entry is added.EXAMPLES:
sage: s = ShiftedPrimedTableau([(1, 2, 3.5, 5, 6.5), (3, 5.5)]) sage: s.pp() 1 2 4' 5 7' 3 6' sage: s.to_chain() [[], 1, [1], 2, [2], 1, [2, 1], 3, [3, 1], 2, [4, 1], 3, [4, 2], 3, [5, 2]] sage: s = ShiftedPrimedTableau([(1, 3.5), (2.5,), (6,)], skew=[2,1]) sage: s.pp() . . 1 4' . 3' 6 sage: s.to_chain() [[2, 1], 2, [3, 1], 0, [3, 1], 3, [3, 2], 3, [4, 2], 0, [4, 2], 1, [4, 2, 1]]
- weight()¶
Return the weight of
self.The weight of a shifted primed tableau is defined to be the vector with \(i\)-th component equal to the number of entries \(i\) and \(i'\) in the tableau.
EXAMPLES:
sage: t = ShiftedPrimedTableau([['2p',2,2],[2,'3p']], skew=[1]) sage: t.weight() (0, 4, 1)
- class sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux(skew=None, primed_diagonal=False)¶
Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.structure.parent.ParentReturns the combinatorial class of shifted primed tableaux subject to the constraints given by the arguments.
A primed tableau is a tableau of shifted shape on the alphabet \(X' = \{1' < 1 < 2' < 2 < \cdots < n' < n\}\) such that
the entries are weakly increasing along rows and columns
a row cannot have two repeated primed entries, and a column cannot have two repeated non-primed entries
INPUT:
Valid optional keywords:
shape– the (outer skew) shape of tableauxweight– the weight of tableauxmax_entry– the maximum entry of tableauxskew– the inner skew shape of tableauxprimed_diagonal– allow primed entries in main diagonal of tableaux
The weight of a tableau is defined to be the vector with \(i\)-th component equal to the number of entries \(i\) and \(i'\) in the tableau. The sum of the coordinates in the weight vector must be equal to the number of entries in the partition.
The
shapeandskewmust be strictly decreasing partitions. Theprimed_diagonalis a boolean (default:False).EXAMPLES:
sage: SPT = ShiftedPrimedTableaux(weight=(1,2,2), shape=[3,2]); SPT Shifted Primed Tableaux of weight (1, 2, 2) and shape [3, 2] sage: SPT.list() [[(1, 2, 2), (3, 3)], [(1, 2', 3'), (2, 3)], [(1, 2', 3'), (2, 3')], [(1, 2', 2), (3, 3)]] sage: SPT = ShiftedPrimedTableaux(weight=(1,2,2), shape=[3,2], ....: primed_diagonal=True); SPT Shifted Primed Tableaux of weight (1, 2, 2) and shape [3, 2] sage: SPT.list() [[(1, 2, 2), (3, 3)], [(1, 2, 2), (3', 3)], [(1, 2', 3'), (2, 3)], [(1, 2', 3'), (2, 3')], [(1, 2', 3'), (2', 3)], [(1, 2', 3'), (2', 3')], [(1, 2', 2), (3, 3)], [(1, 2', 2), (3', 3)], [(1', 2, 2), (3, 3)], [(1', 2, 2), (3', 3)], [(1', 2', 3'), (2, 3)], [(1', 2', 3'), (2, 3')], [(1', 2', 3'), (2', 3)], [(1', 2', 3'), (2', 3')], [(1', 2', 2), (3, 3)], [(1', 2', 2), (3', 3)]] sage: SPT = ShiftedPrimedTableaux(weight=(1,2)); SPT Shifted Primed Tableaux of weight (1, 2) sage: list(SPT) [[(1, 2, 2)], [(1, 2', 2)], [(1, 2'), (2,)]] sage: SPT = ShiftedPrimedTableaux(weight=(1,2), primed_diagonal=True) sage: list(SPT) [[(1, 2, 2)], [(1, 2', 2)], [(1', 2, 2)], [(1', 2', 2)], [(1, 2'), (2,)], [(1, 2'), (2',)], [(1', 2'), (2,)], [(1', 2'), (2',)]] sage: SPT = ShiftedPrimedTableaux([3,2], max_entry=2); SPT Shifted Primed Tableaux of shape [3, 2] and maximum entry 2 sage: list(SPT) [[(1, 1, 1), (2, 2)], [(1, 1, 2'), (2, 2)]] sage: SPT = ShiftedPrimedTableaux([3,2], max_entry=2, ....: primed_diagonal=True) sage: list(SPT) [[(1, 1, 1), (2, 2)], [(1, 1, 1), (2', 2)], [(1', 1, 1), (2, 2)], [(1', 1, 1), (2', 2)], [(1, 1, 2'), (2, 2)], [(1, 1, 2'), (2', 2)], [(1', 1, 2'), (2, 2)], [(1', 1, 2'), (2', 2)]]
See also
- Element¶
alias of
ShiftedPrimedTableau
- options(*get_value, **set_value)¶
Sets the global options for elements of the tableau, skew_tableau, and tableau tuple classes. The defaults are for tableau to be displayed as a list, latexed as a Young diagram using the English convention.
OPTIONS:
ascii_art– (default:repr) Controls the ascii art output for tableauxcompact– minimal length ascii artrepr– display using the diagram string representationtable– display as a table
convention– (default:English) Sets the convention used for displaying tableaux and partitionsEnglish– use the English conventionFrench– use the French convention
display– (default:list) Controls the way in which tableaux are printedarray– alias fordiagramcompact– minimal length string representationdiagram– display as Young diagram (similar topp()ferrers_diagram– alias fordiagramlist– print tableaux as listsyoung_diagram– alias fordiagram
latex– (default:diagram) Controls the way in which tableaux are latexedarray– alias fordiagramdiagram– as a Young diagramferrers_diagram– alias fordiagramlist– as a listyoung_diagram– alias fordiagram
notation– alternative name forconvention
Note
Changing the
conventionfor tableaux also changes theconventionfor partitions.If no parameters are set, then the function returns a copy of the options dictionary.
EXAMPLES:
sage: T = Tableau([[1,2,3],[4,5]]) sage: T [[1, 2, 3], [4, 5]] sage: Tableaux.options.display="array" sage: T 1 2 3 4 5 sage: Tableaux.options.convention="french" sage: T 4 5 1 2 3
Changing the
conventionfor tableaux also changes theconventionfor partitions and vice versa:sage: P = Partition([3,3,1]) sage: print(P.ferrers_diagram()) * *** *** sage: Partitions.options.convention="english" sage: print(P.ferrers_diagram()) *** *** * sage: T 1 2 3 4 5
The ASCII art can also be changed:
sage: t = Tableau([[1,2,3],[4,5]]) sage: ascii_art(t) 1 2 3 4 5 sage: Tableaux.options.ascii_art = "table" sage: ascii_art(t) +---+---+---+ | 1 | 2 | 3 | +---+---+---+ | 4 | 5 | +---+---+ sage: Tableaux.options.ascii_art = "compact" sage: ascii_art(t) |1|2|3| |4|5| sage: Tableaux.options._reset()
See
GlobalOptionsfor more features of these options.
- class sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_all(skew=None, primed_diagonal=False)¶
Bases:
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableauxThe class of all shifted primed tableaux.
- class sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_shape(shape, max_entry=None, skew=None, primed_diagonal=False)¶
Bases:
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableauxShifted primed tableaux of a fixed shape.
Shifted primed tableaux admit a type \(A_n\) classical crystal structure with highest weights corresponding to a given shape.
The list of module generators consists of all elements of the crystal with nonincreasing weight entries.
The crystal is constructed following operations described in [HPS2017] and [AO2018].
The optional
primed_diagonalallows primed entries in the main diagonal of all the Shifted primed tableaux of a fixed shape. If themax_entryisNonethenmax_entryis set to the total number of entries in the tableau ifprimed_diagonalisTrue.EXAMPLES:
sage: ShiftedPrimedTableaux([4,3,1], max_entry=4) Shifted Primed Tableaux of shape [4, 3, 1] and maximum entry 4 sage: ShiftedPrimedTableaux([4,3,1], max_entry=4).cardinality() 384
We compute some of the crystal structure:
sage: SPTC = crystals.ShiftedPrimedTableaux([3,2], 3) sage: T = SPTC.module_generators[-1] sage: T [(1, 1, 2'), (2, 3')] sage: T.f(2) [(1, 1, 3'), (2, 3')] sage: len(SPTC.module_generators) 7 sage: SPTC[0] [(1, 1, 1), (2, 2)] sage: SPTC.cardinality() 24
We compare this implementation with the \(q(n)\)-crystal on (tensor products) of letters:
sage: tableau_crystal = crystals.ShiftedPrimedTableaux([4,1], 3) sage: tableau_digraph = tableau_crystal.digraph() sage: c = crystals.Letters(['Q', 3]) sage: tensor_crystal = tensor([c]*5) sage: u = tensor_crystal(c(1), c(1), c(1), c(2), c(1)) sage: subcrystal = tensor_crystal.subcrystal(generators=[u], ....: index_set=[1,2,-1]) sage: tensor_digraph = subcrystal.digraph() sage: tensor_digraph.is_isomorphic(tableau_digraph, edge_labels=True) True
If we allow primed entries in the main diagonal:
sage: ShiftedPrimedTableaux([4,3,1], max_entry=4, ....: primed_diagonal=True) Shifted Primed Tableaux of shape [4, 3, 1] and maximum entry 4 sage: ShiftedPrimedTableaux([4,3,1], max_entry=4, ....: primed_diagonal=True).cardinality() 3072 sage: SPTC = ShiftedPrimedTableaux([3,2], max_entry=3, ....: primed_diagonal=True) sage: T = SPTC[-1] sage: T [(1', 2', 2), (3', 3)] sage: SPTC[0] [(1, 1, 1), (2, 2)] sage: SPTC.cardinality() 96
- module_generators()¶
Return the generators of
selfas a crystal.
- shape()¶
Return the shape of the shifted tableaux
self.
- class sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_weight(weight, skew=None, primed_diagonal=False)¶
Bases:
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableauxShifted primed tableaux of fixed weight.
EXAMPLES:
sage: ShiftedPrimedTableaux(weight=(2,3,1)) Shifted Primed Tableaux of weight (2, 3, 1) sage: ShiftedPrimedTableaux(weight=(2,3,1)).cardinality() 17 sage: SPT = ShiftedPrimedTableaux(weight=(2,3,1), primed_diagonal=True) sage: SPT.cardinality() 64 sage: T = ShiftedPrimedTableaux(weight=(3,2), primed_diagonal=True) sage: T[:5] [[(1, 1, 1, 2, 2)], [(1, 1, 1, 2', 2)], [(1', 1, 1, 2, 2)], [(1', 1, 1, 2', 2)], [(1, 1, 1, 2), (2,)]] sage: T.cardinality() 16
- class sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_weight_shape(weight, shape, skew=None, primed_diagonal=False)¶
Bases:
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableauxShifted primed tableaux of the fixed weight and shape.
EXAMPLES:
sage: ShiftedPrimedTableaux([4,2,1], weight=(2,3,2)) Shifted Primed Tableaux of weight (2, 3, 2) and shape [4, 2, 1] sage: ShiftedPrimedTableaux([4,2,1], weight=(2,3,2)).cardinality() 4 sage: T = ShiftedPrimedTableaux([4,2,1], weight=(2,3,2), ....: primed_diagonal=True) sage: T[:6] [[(1, 1, 2, 2), (2, 3'), (3,)], [(1, 1, 2, 2), (2, 3'), (3',)], [(1, 1, 2, 2), (2', 3'), (3,)], [(1, 1, 2, 2), (2', 3'), (3',)], [(1, 1, 2', 3), (2, 2), (3,)], [(1, 1, 2', 3), (2, 2), (3',)]] sage: T.cardinality() 32