marked_partition

marked_partition#

Marked partition

The suspensions of interval exchanges have a natural geometric structure which is a translation surface. The singularities of the interval exchanges yield singularities on the surface which gives an integer partition. As the left and the right of the interval plays a special role, we consider partitions with marking. Either the points at the left and the right of the interval coincide in which case the marking is of type one, otherwise they are two different parts and the marking is of type two.

class surface_dynamics.interval_exchanges.marked_partition.MarkedPartition(*args, **kwds)#

Marked partition

A marked partition is an integer partition p with the additional data of either a couple (m,a) where m is an element of p and a is an integer lesser than p or a couple (m_l,m_r) where m_l and m_r are distinct element of p. In the first case the type is 1 while in the second the type is 2.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import MarkedPartition
sage: m1 = MarkedPartition([3,2,1],1,(2,0)); m1
2|0 [3, 2, 1]
sage: m2 = MarkedPartition([3,2,1],2,(2,3)); m2
2o3 [3, 2, 1]

It also possible to initialize it from a string:

sage: MarkedPartition('2|1 [3,2,2]')
2|1 [3, 2, 2]

TESTS:

sage: MarkedPartition(m1)
2|0 [3, 2, 1]
sage: MarkedPartition(str(m1))
2|0 [3, 2, 1]
sage: MarkedPartition(m2)
2o3 [3, 2, 1]
sage: MarkedPartition(str(m2))
2o3 [3, 2, 1]
is_odd()#

Return True if all terms of p are odd.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import MarkedPartition
sage: MarkedPartition([3,1,1],1,(3,1)).is_odd()
True
sage: MarkedPartition([3,2,2],1,(3,1)).is_odd()
False
left()#

Return the part that is marked on the left.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import MarkedPartition
sage: MarkedPartition([3,2,1],1,(2,0)).left()
2
sage: MarkedPartition([3,2,1],2,(1,3)).left()
1
marking()#

Return a 3-tuple (type,d0,d1) where type is the type of the marking and (d0,d1) corresponds to the data associated to the marking.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import MarkedPartition
sage: MarkedPartition([3,2,1],1,(2,0)).marking()
2|0
partition()#

Return the underlying partition.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import MarkedPartition
sage: MarkedPartition([1,3,2],1,(2,0)).partition()
[3, 2, 1]
sage: MarkedPartition([2,3,1],2,(1,3)).partition()
[3, 2, 1]
right()#

Return the part that is marked on the right.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import MarkedPartition
sage: MarkedPartition([3,2,1],1,(2,0)).right()
2
sage: MarkedPartition([3,2,1],2,(1,3)).right()
3
class surface_dynamics.interval_exchanges.marked_partition.Marking(*args)#

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import Marking
sage: Marking('2o3')
2o3
sage: Marking('3|0')
3|0
sage: Marking(1,(2,0))
2|0
sage: Marking(2,(2,1))
2o1
left()#

Return the part that is marked on the left.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import Marking
sage: Marking(1,(2,0)).left()
2
sage: Marking(2,(1,3)).left()
1
right()#

Return the part that is marked on the right.

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import Marking
sage: Marking(1,(2,0)).right()
2
sage: Marking(2,(1,3)).right()
3
surface_dynamics.interval_exchanges.marked_partition.markings(p)#

Return all possible markings attached to a given partition.

INPUT:

  • p - a list of non-negative integers

EXAMPLES:

sage: from surface_dynamics.interval_exchanges.marked_partition import markings
sage: list(markings([2,1]))
[2|0, 2|1, 2|2, 1|0, 1|1, 2o1, 1o2]
sage: list(markings([2,2,1]))
[2|0, 2|1, 2|2, 1|0, 1|1, 2o2, 2o1, 1o2]
sage: list(markings([2,2,1,1]))
[2|0, 2|1, 2|2, 1|0, 1|1, 2o2, 2o1, 1o2, 1o1]
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