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One way to do that is using the conjugate partition.

For any partition, the conjugate partition is the one whose lines are the columns of p; analogue to the transpose for matrices.

Here is how to use it to remove vertical border strip.

sage: p = Partition([5,3,1])
sage: p
[5, 3, 1]
sage: unicode_art(p)
┌┬┬┬┬┐
├┼┼┼┴┘
├┼┴┘
└┘
sage: h = p.remove_horizontal_border_strip(1).list()
sage: h
[[5, 3], [5, 2, 1], [4, 3, 1]]
sage: unicode_art(q)
⎡         ┌┬┬┬┬┐  ┌┬┬┬┐ ⎤
⎢ ┌┬┬┬┬┐  ├┼┼┴┴┘  ├┼┼┼┘ ⎥
⎢ ├┼┼┼┴┘  ├┼┘     ├┼┴┘  ⎥
⎣ └┴┴┘  , └┘    , └┘    ⎦
sage: pc = p.conjugate()
sage: pc
[3, 2, 2, 1, 1]
sage: unicode_art(pc)
┌┬┬┐
├┼┼┘
├┼┤
├┼┘
├┤
└┘
sage: qq = [q.conjugate() for q in pc.remove_horizontal_border_strip(1)]
sage: qq
[[4, 3, 1], [5, 2, 1], [5, 3]]
sage: unicode_art(qq)
⎡ ┌┬┬┬┐  ┌┬┬┬┬┐         ⎤
⎢ ├┼┼┼┘  ├┼┼┴┴┘  ┌┬┬┬┬┐ ⎥
⎢ ├┼┴┘   ├┼┘     ├┼┼┼┴┘ ⎥
⎣ └┘   , └┘    , └┴┴┘   ⎦


To define a function to do it, just do:

def remove_vertical_border_strip(p, k):
pc = p.conjugate()
return [q.conjugate() for q in pc.remove_horizontal_border_strip(k)]


and then you can just do:

sage: p = Partition([5, 3, 1])
sage: qq = remove_vertical_border_strip(p, 1)
sage: unicode_art(qq)
⎡ ┌┬┬┬┐  ┌┬┬┬┬┐         ⎤
⎢ ├┼┼┼┘  ├┼┼┴┴┘  ┌┬┬┬┬┐ ⎥
⎢ ├┼┴┘   ├┼┘     ├┼┼┼┴┘ ⎥
⎣ └┘   , └┘    , └┴┴┘   ⎦


One way to do that is using the conjugate partition.

For any partition, the conjugate partition is the one whose lines are the columns of p; analogue to the transpose for matrices.

Here is how to use it to remove vertical border strip.

sage: p = Partition([5,3,1])
sage: p
[5, 3, 1]
sage: unicode_art(p)
┌┬┬┬┬┐
├┼┼┼┴┘
├┼┴┘
└┘
sage: h = p.remove_horizontal_border_strip(1).list()
sage: h
[[5, 3], [5, 2, 1], [4, 3, 1]]
sage: unicode_art(q)
⎡         ┌┬┬┬┬┐  ┌┬┬┬┐ ⎤
⎢ ┌┬┬┬┬┐  ├┼┼┴┴┘  ├┼┼┼┘ ⎥
⎢ ├┼┼┼┴┘  ├┼┘     ├┼┴┘  ⎥
⎣ └┴┴┘  , └┘    , └┘    ⎦
sage: pc = p.conjugate()
sage: pc
[3, 2, 2, 1, 1]
sage: unicode_art(pc)
┌┬┬┐
├┼┼┘
├┼┤
├┼┘
├┤
└┘
sage: qq = [q.conjugate() for q in pc.remove_horizontal_border_strip(1)]
sage: qq
[[4, 3, 1], [5, 2, 1], [5, 3]]
sage: unicode_art(qq)
⎡ ┌┬┬┬┐  ┌┬┬┬┬┐         ⎤
⎢ ├┼┼┼┘  ├┼┼┴┴┘  ┌┬┬┬┬┐ ⎥
⎢ ├┼┴┘   ├┼┘     ├┼┼┼┴┘ ⎥
⎣ └┘   , └┘    , └┴┴┘   ⎦


To One can define a function to do it, just do:corresponding function:

def remove_vertical_border_strip(p, k):
r"""
Return the partitions obtained from p by removing
a vertical border strip of length k.

EXAMPLE::

sage: p = Partition([5, 3, 1])
sage: remove_vertical_border_strip(p, 1)
[[4, 3, 1], [5, 2, 1], [5, 3]]
"""
pc = p.conjugate()
return [q.conjugate() for q in pc.remove_horizontal_border_strip(k)]


and then you can just do:use that function:

sage: p = Partition([5, 3, 1])
sage: qq = remove_vertical_border_strip(p, 1)
sage: qq
[[4, 3, 1], [5, 2, 1], [5, 3]]
sage: unicode_art(qq)
⎡ ┌┬┬┬┐  ┌┬┬┬┬┐         ⎤
⎢ ├┼┼┼┘  ├┼┼┴┴┘  ┌┬┬┬┬┐ ⎥
⎢ ├┼┴┘   ├┼┘     ├┼┼┼┴┘ ⎥
⎣ └┘   , └┘    , └┴┴┘   ⎦