Ribbon Shaped Tableaux

sage.combinat.ribbon_shaped_tableau.RibbonShapedTableau

A ribbon shaped tableau.

For the purposes of this class, a ribbon shaped tableau is a skew tableau whose shape is a skew partition which:

  • has at least one cell in row \(1\);
  • has at least one cell in column \(1\);
  • has exactly one cell in each of \(q\) consecutive diagonals, for some nonnegative integer \(q\).

A ribbon is given by a list of the rows from top to bottom.

EXAMPLES:

sage: x = RibbonShapedTableau([[None, None, None, 2, 3], [None, 1, 4, 5], [3, 2]]); x
[[None, None, None, 2, 3], [None, 1, 4, 5], [3, 2]]
sage: x.pp()
  .  .  .  2  3
  .  1  4  5
  3  2
sage: x.shape()
[5, 4, 2] / [3, 1]

The entries labeled by None correspond to the inner partition. Using None is optional; the entries will be shifted accordingly.

sage: x = RibbonShapedTableau([[2,3],[1,4,5],[3,2]]); x.pp()
  .  .  .  2  3
  .  1  4  5
  3  2
sage.combinat.ribbon_shaped_tableau.RibbonShapedTableaux

The set of all ribbon shaped tableaux.

sage.combinat.ribbon_shaped_tableau.Ribbon_class

This exists solely for unpickling Ribbon_class objects.

sage.combinat.ribbon_shaped_tableau.StandardRibbonShapedTableaux

The set of all standard ribbon shaped tableaux.

INPUT:

  • shape – (optional) the composition shape of the rows
sage.combinat.ribbon_shaped_tableau.StandardRibbonShapedTableaux_shape

Class of standard ribbon shaped tableaux of ribbon shape shape.

EXAMPLES:

sage: StandardRibbonShapedTableaux([2,2])
Standard ribbon shaped tableaux of shape [2, 2]
sage: StandardRibbonShapedTableaux([2,2]).first()
[[None, 2, 4], [1, 3]]
sage: StandardRibbonShapedTableaux([2,2]).last()
[[None, 1, 2], [3, 4]]
sage: StandardRibbonShapedTableaux([2,2]).cardinality()
5
sage: StandardRibbonShapedTableaux([2,2]).list()
[[[None, 1, 3], [2, 4]],
 [[None, 1, 2], [3, 4]],
 [[None, 2, 3], [1, 4]],
 [[None, 2, 4], [1, 3]],
 [[None, 1, 4], [2, 3]]]
sage: StandardRibbonShapedTableaux([3,2,2]).cardinality()
155