Output functions¶

These are the output functions for latexing and ascii/unicode art versions of partitions and tableaux.

AUTHORS:

Mike Hansen (?): initial version
Andrew Mathas (2013-02-14): Added support for displaying conventions and lines, and tableaux of skew partition, composition, and skew/composition/partition/tableaux tuple shape.
Travis Scrimshaw (2020-08): Added support for ascii/unicode art

sage.combinat.output.ascii_art_table(data, use_unicode=False, convention='English')[source]¶

Return an ascii art table of data.

EXAMPLES:

Sage

sage: from sage.combinat.output import ascii_art_table

sage: data = [[None, None, 1], [2, 2], [3,4,5], [None, None, 10], [], [6]]
sage: print(ascii_art_table(data))
        +----+
        | 1  |
+---+---+----+
| 2 | 2 |
+---+---+----+
| 3 | 4 | 5  |
+---+---+----+
        | 10 |
        +----+

+---+
| 6 |
+---+
sage: print(ascii_art_table(data, use_unicode=True))
        ┌────┐
        │ 1  │
┌───┬───┼────┘
│ 2 │ 2 │
├───┼───┼────┐
│ 3 │ 4 │ 5  │
└───┴───┼────┤
        │ 10 │
        └────┘

┌───┐
│ 6 │
└───┘

sage: data = [[1, None, 2], [None, 2]]
sage: print(ascii_art_table(data))
+---+   +---+
| 1 |   | 2 |
+---+---+---+
    | 2 |
    +---+
sage: print(ascii_art_table(data, use_unicode=True))
┌───┐   ┌───┐
│ 1 │   │ 2 │
└───┼───┼───┘
    │ 2 │
    └───┘

Python

>>> from sage.all import *
>>> from sage.combinat.output import ascii_art_table

>>> data = [[None, None, Integer(1)], [Integer(2), Integer(2)], [Integer(3),Integer(4),Integer(5)], [None, None, Integer(10)], [], [Integer(6)]]
>>> print(ascii_art_table(data))
        +----+
        | 1  |
+---+---+----+
| 2 | 2 |
+---+---+----+
| 3 | 4 | 5  |
+---+---+----+
        | 10 |
        +----+
<BLANKLINE>
+---+
| 6 |
+---+
>>> print(ascii_art_table(data, use_unicode=True))
        ┌────┐
        │ 1  │
┌───┬───┼────┘
│ 2 │ 2 │
├───┼───┼────┐
│ 3 │ 4 │ 5  │
└───┴───┼────┤
        │ 10 │
        └────┘
<BLANKLINE>
┌───┐
│ 6 │
└───┘

>>> data = [[Integer(1), None, Integer(2)], [None, Integer(2)]]
>>> print(ascii_art_table(data))
+---+   +---+
| 1 |   | 2 |
+---+---+---+
    | 2 |
    +---+
>>> print(ascii_art_table(data, use_unicode=True))
┌───┐   ┌───┐
│ 1 │   │ 2 │
└───┼───┼───┘
    │ 2 │
    └───┘

sage.combinat.output.ascii_art_table_russian(data, use_unicode=False, compact=False)[source]¶

Return an ascii art table of data for the russian convention.

EXAMPLES:

Sage

sage: from sage.combinat.output import ascii_art_table_russian
sage: data = [[None, None, 1], [2, 2], [3,4,5], [None, None, 10], [], [6]]
sage: print(ascii_art_table_russian(data))
   / \         / \
  /   \       /   \
 \  6  /     \ 10  \
  \   /       \   / \
   \ /         \ /   \
                X  5  /
               / \   /
              /   \ /
             /  4  X
            / \   / \   / \
           /   \ /   \ /   \
          \  3  X  2  X  1  /
           \   / \   / \   /
            \ /   \ /   \ /
             \  2  /
              \   /
               \ /
sage: print(ascii_art_table_russian(data, use_unicode=True))
   ╱ ╲         ╱ ╲
  ╱   ╲       ╱   ╲
 ╲  6  ╱     ╲ 10  ╲
  ╲   ╱       ╲   ╱ ╲
   ╲ ╱         ╲ ╱   ╲
                ╳  5  ╱
               ╱ ╲   ╱
              ╱   ╲ ╱
             ╱  4  ╳
            ╱ ╲   ╱ ╲   ╱ ╲
           ╱   ╲ ╱   ╲ ╱   ╲
          ╲  3  ╳  2  ╳  1  ╱
           ╲   ╱ ╲   ╱ ╲   ╱
            ╲ ╱   ╲ ╱   ╲ ╱
             ╲  2  ╱
              ╲   ╱
               ╲ ╱
sage: data = [[1, None, 2], [None, 2]]
sage: print(ascii_art_table_russian(data))
  / \ / \
 \ 2 X 2 /
  \ / \ /
   X
  / \
 \ 1 /
  \ /
sage: print(ascii_art_table_russian(data, use_unicode=True))
  ╱ ╲ ╱ ╲
 ╲ 2 ╳ 2 ╱
  ╲ ╱ ╲ ╱
   ╳
  ╱ ╲
 ╲ 1 ╱
  ╲ ╱

Python

>>> from sage.all import *
>>> from sage.combinat.output import ascii_art_table_russian
>>> data = [[None, None, Integer(1)], [Integer(2), Integer(2)], [Integer(3),Integer(4),Integer(5)], [None, None, Integer(10)], [], [Integer(6)]]
>>> print(ascii_art_table_russian(data))
   / \         / \
  /   \       /   \
 \  6  /     \ 10  \
  \   /       \   / \
   \ /         \ /   \
                X  5  /
               / \   /
              /   \ /
             /  4  X
            / \   / \   / \
           /   \ /   \ /   \
          \  3  X  2  X  1  /
           \   / \   / \   /
            \ /   \ /   \ /
             \  2  /
              \   /
               \ /
>>> print(ascii_art_table_russian(data, use_unicode=True))
   ╱ ╲         ╱ ╲
  ╱   ╲       ╱   ╲
 ╲  6  ╱     ╲ 10  ╲
  ╲   ╱       ╲   ╱ ╲
   ╲ ╱         ╲ ╱   ╲
                ╳  5  ╱
               ╱ ╲   ╱
              ╱   ╲ ╱
             ╱  4  ╳
            ╱ ╲   ╱ ╲   ╱ ╲
           ╱   ╲ ╱   ╲ ╱   ╲
          ╲  3  ╳  2  ╳  1  ╱
           ╲   ╱ ╲   ╱ ╲   ╱
            ╲ ╱   ╲ ╱   ╲ ╱
             ╲  2  ╱
              ╲   ╱
               ╲ ╱
>>> data = [[Integer(1), None, Integer(2)], [None, Integer(2)]]
>>> print(ascii_art_table_russian(data))
  / \ / \
 \ 2 X 2 /
  \ / \ /
   X
  / \
 \ 1 /
  \ /
>>> print(ascii_art_table_russian(data, use_unicode=True))
  ╱ ╲ ╱ ╲
 ╲ 2 ╳ 2 ╱
  ╲ ╱ ╲ ╱
   ╳
  ╱ ╲
 ╲ 1 ╱
  ╲ ╱

sage.combinat.output.box_exists(tab, i, j)[source]¶

Return True if tab[i][j] exists and is not None; in particular this allows for \(tab[i][j]\) to be '' or 0.

INPUT:

tab – list of lists
i – first coordinate
j – second coordinate

sage.combinat.output.tex_from_array(array, with_lines=True)[source]¶

Return a latex string for a two dimensional array of partition, composition or skew composition shape.

INPUT:

array – list of list
with_lines – boolean (default: True); whether to draw a line to separate the entries in the array

Empty rows are allowed; however, such rows should be given as [None] rather than [].

The array is drawn using either the English or French convention following Tableaux.options().

See also

tex_from_array_tuple()

EXAMPLES:

Sage

sage: from sage.combinat.output import tex_from_array
sage: print(tex_from_array([[1,2,3],[4,5]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\lr{4}&\lr{5}\\\cline{1-2}
\end{array}$}
}
sage: print(tex_from_array([[1,2,3],[4,5]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\\
\lr{1}&\lr{2}&\lr{3}\\
\lr{4}&\lr{5}\\
\end{array}$}
}
sage: print(tex_from_array([[1,2,3],[4,5,6,7],[8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-4}
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\\cline{1-4}
\lr{8}\\\cline{1-1}
\end{array}$}
}
sage: print(tex_from_array([[1,2,3],[4,5,6,7],[8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
\lr{1}&\lr{2}&\lr{3}\\
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\
\lr{8}\\
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\cline{3-3}
&&\lr{3}\\\cline{2-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{1-4}
\lr{8}\\\cline{1-1}
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[None,8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\cline{3-3}
&&\lr{3}\\\cline{2-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{2-4}
&\lr{8}\\\cline{2-2}
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
&&\lr{3}\\
&\lr{5}&\lr{6}&\lr{7}\\
\lr{8}\\
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[None,8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
&&\lr{3}\\
&\lr{5}&\lr{6}&\lr{7}\\
&\lr{8}\\
\end{array}$}
}
sage: Tableaux.options.convention="french"
sage: print(tex_from_array([[1,2,3],[4,5]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{4}&\lr{5}\\\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\end{array}$}
}
sage: print(tex_from_array([[1,2,3],[4,5]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\\
\lr{4}&\lr{5}\\
\lr{1}&\lr{2}&\lr{3}\\
\end{array}$}
}
sage: print(tex_from_array([[1,2,3],[4,5,6,7],[8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{8}\\\cline{1-4}
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\\cline{1-4}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\end{array}$}
}
sage: print(tex_from_array([[1,2,3],[4,5,6,7],[8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\\
\lr{8}\\
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\
\lr{1}&\lr{2}&\lr{3}\\
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{8}\\\cline{1-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{2-4}
&&\lr{3}\\\cline{3-3}
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[None,8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\cline{2-2}
&\lr{8}\\\cline{2-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{2-4}
&&\lr{3}\\\cline{3-3}
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\\
\lr{8}\\
&\lr{5}&\lr{6}&\lr{7}\\
&&\lr{3}\\
\end{array}$}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[None,8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\\
&\lr{8}\\
&\lr{5}&\lr{6}&\lr{7}\\
&&\lr{3}\\
\end{array}$}
}
sage: Tableaux.options.convention="russian"
sage: print(tex_from_array([[1,2,3],[4,5]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\\cline{1-3}
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\\cline{1-3}
\end{array}$}}
}
sage: print(tex_from_array([[1,2,3],[4,5]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\\
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}
sage: print(tex_from_array([[1,2,3],[4,5,6,7],[8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{\rotatebox{-45}{8}}\\\cline{1-4}
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{1-4}
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\\cline{1-3}
\end{array}$}}
}
sage: print(tex_from_array([[1,2,3],[4,5,6,7],[8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\\
\lr{\rotatebox{-45}{8}}\\
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{\rotatebox{-45}{8}}\\\cline{1-4}
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{2-4}
&&\lr{\rotatebox{-45}{3}}\\\cline{3-3}
\end{array}$}}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[None,8]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\cline{2-2}
&\lr{\rotatebox{-45}{8}}\\\cline{2-4}
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{2-4}
&&\lr{\rotatebox{-45}{3}}\\\cline{3-3}
\end{array}$}}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\\
\lr{\rotatebox{-45}{8}}\\
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
&&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}
sage: print(tex_from_array([[None,None,3],[None,5,6,7],[None,8]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\\
&\lr{\rotatebox{-45}{8}}\\
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
&&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}

sage: Tableaux.options._reset()

Python

>>> from sage.all import *
>>> from sage.combinat.output import tex_from_array
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\lr{4}&\lr{5}\\\cline{1-2}
\end{array}$}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\\
\lr{1}&\lr{2}&\lr{3}\\
\lr{4}&\lr{5}\\
\end{array}$}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5),Integer(6),Integer(7)],[Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-4}
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\\cline{1-4}
\lr{8}\\\cline{1-1}
\end{array}$}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5),Integer(6),Integer(7)],[Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
\lr{1}&\lr{2}&\lr{3}\\
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\
\lr{8}\\
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\cline{3-3}
&&\lr{3}\\\cline{2-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{1-4}
\lr{8}\\\cline{1-1}
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[None,Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\cline{3-3}
&&\lr{3}\\\cline{2-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{2-4}
&\lr{8}\\\cline{2-2}
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
&&\lr{3}\\
&\lr{5}&\lr{6}&\lr{7}\\
\lr{8}\\
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[None,Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
&&\lr{3}\\
&\lr{5}&\lr{6}&\lr{7}\\
&\lr{8}\\
\end{array}$}
}
>>> Tableaux.options.convention="french"
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{4}&\lr{5}\\\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\end{array}$}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\\
\lr{4}&\lr{5}\\
\lr{1}&\lr{2}&\lr{3}\\
\end{array}$}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5),Integer(6),Integer(7)],[Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{8}\\\cline{1-4}
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\\cline{1-4}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\end{array}$}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5),Integer(6),Integer(7)],[Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\\
\lr{8}\\
\lr{4}&\lr{5}&\lr{6}&\lr{7}\\
\lr{1}&\lr{2}&\lr{3}\\
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{8}\\\cline{1-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{2-4}
&&\lr{3}\\\cline{3-3}
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[None,Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\cline{2-2}
&\lr{8}\\\cline{2-4}
&\lr{5}&\lr{6}&\lr{7}\\\cline{2-4}
&&\lr{3}\\\cline{3-3}
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\\
\lr{8}\\
&\lr{5}&\lr{6}&\lr{7}\\
&&\lr{3}\\
\end{array}$}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[None,Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{4}c}\\
&\lr{8}\\
&\lr{5}&\lr{6}&\lr{7}\\
&&\lr{3}\\
\end{array}$}
}
>>> Tableaux.options.convention="russian"
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\\cline{1-3}
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\\cline{1-3}
\end{array}$}}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\\
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5),Integer(6),Integer(7)],[Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{\rotatebox{-45}{8}}\\\cline{1-4}
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{1-4}
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\\cline{1-3}
\end{array}$}}
}
>>> print(tex_from_array([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5),Integer(6),Integer(7)],[Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\\
\lr{\rotatebox{-45}{8}}\\
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\cline{1-1}
\lr{\rotatebox{-45}{8}}\\\cline{1-4}
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{2-4}
&&\lr{\rotatebox{-45}{3}}\\\cline{3-3}
\end{array}$}}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[None,Integer(8)]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\cline{2-2}
&\lr{\rotatebox{-45}{8}}\\\cline{2-4}
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{2-4}
&&\lr{\rotatebox{-45}{3}}\\\cline{3-3}
\end{array}$}}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\\
\lr{\rotatebox{-45}{8}}\\
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
&&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}
>>> print(tex_from_array([[None,None,Integer(3)],[None,Integer(5),Integer(6),Integer(7)],[None,Integer(8)]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{4}c}\\
&\lr{\rotatebox{-45}{8}}\\
&\lr{\rotatebox{-45}{5}}&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
&&\lr{\rotatebox{-45}{3}}\\
\end{array}$}}
}

>>> Tableaux.options._reset()

sage.combinat.output.tex_from_array_tuple(a_tuple, with_lines=True)[source]¶

Return a latex string for a tuple of two dimensional array of partition, composition or skew composition shape.

INPUT:

a_tuple – tuple of lists of lists
with_lines – boolean (default: True); whether to draw lines to separate the entries in the components of a_tuple

See also

tex_from_array() for the description of each array

EXAMPLES:

Sage

sage: from sage.combinat.output import tex_from_array_tuple
sage: print(tex_from_array_tuple([[[1,2,3],[4,5]],[],[[None,6,7],[None,8],[9]]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\lr{4}&\lr{5}\\\cline{1-2}
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{2-3}
&\lr{6}&\lr{7}\\\cline{2-3}
&\lr{8}\\\cline{1-2}
\lr{9}\\\cline{1-1}
\end{array}$}
}
sage: print(tex_from_array_tuple([[[1,2,3],[4,5]],[],[[None,6,7],[None,8],[9]]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\\
\lr{1}&\lr{2}&\lr{3}\\
\lr{4}&\lr{5}\\
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\\
&\lr{6}&\lr{7}\\
&\lr{8}\\
\lr{9}\\
\end{array}$}
}
sage: Tableaux.options.convention="french"
sage: print(tex_from_array_tuple([[[1,2,3],[4,5]],[],[[None,6,7],[None,8],[9]]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{4}&\lr{5}\\\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\cline{1-1}
\lr{9}\\\cline{1-2}
&\lr{8}\\\cline{2-3}
&\lr{6}&\lr{7}\\\cline{2-3}
\end{array}$}
}
sage: print(tex_from_array_tuple([[[1,2,3],[4,5]],[],[[None,6,7],[None,8],[9]]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\\
\lr{4}&\lr{5}\\
\lr{1}&\lr{2}&\lr{3}\\
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\\
\lr{9}\\
&\lr{8}\\
&\lr{6}&\lr{7}\\
\end{array}$}
}
sage: Tableaux.options.convention="russian"
sage: print(tex_from_array_tuple([[[1,2,3],[4,5]],[],[[None,6,7],[None,8],[9]]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\\cline{1-3}
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\\cline{1-3}
\end{array}$}},\emptyset,\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\cline{1-1}
\lr{\rotatebox{-45}{9}}\\\cline{1-2}
&\lr{\rotatebox{-45}{8}}\\\cline{2-3}
&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{2-3}
\end{array}$}}
}
sage: print(tex_from_array_tuple([[[1,2,3],[4,5]],[],[[None,6,7],[None,8],[9]]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\\
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\
\end{array}$}},\emptyset,\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\\
\lr{\rotatebox{-45}{9}}\\
&\lr{\rotatebox{-45}{8}}\\
&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
\end{array}$}}
}

sage: Tableaux.options._reset()

Python

>>> from sage.all import *
>>> from sage.combinat.output import tex_from_array_tuple
>>> print(tex_from_array_tuple([[[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]],[],[[None,Integer(6),Integer(7)],[None,Integer(8)],[Integer(9)]]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\lr{4}&\lr{5}\\\cline{1-2}
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{2-3}
&\lr{6}&\lr{7}\\\cline{2-3}
&\lr{8}\\\cline{1-2}
\lr{9}\\\cline{1-1}
\end{array}$}
}
>>> print(tex_from_array_tuple([[[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]],[],[[None,Integer(6),Integer(7)],[None,Integer(8)],[Integer(9)]]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\\
\lr{1}&\lr{2}&\lr{3}\\
\lr{4}&\lr{5}\\
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\\
&\lr{6}&\lr{7}\\
&\lr{8}\\
\lr{9}\\
\end{array}$}
}
>>> Tableaux.options.convention="french"
>>> print(tex_from_array_tuple([[[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]],[],[[None,Integer(6),Integer(7)],[None,Integer(8)],[Integer(9)]]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{4}&\lr{5}\\\cline{1-3}
\lr{1}&\lr{2}&\lr{3}\\\cline{1-3}
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\cline{1-1}
\lr{9}\\\cline{1-2}
&\lr{8}\\\cline{2-3}
&\lr{6}&\lr{7}\\\cline{2-3}
\end{array}$}
}
>>> print(tex_from_array_tuple([[[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]],[],[[None,Integer(6),Integer(7)],[None,Integer(8)],[Integer(9)]]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\\
\lr{4}&\lr{5}\\
\lr{1}&\lr{2}&\lr{3}\\
\end{array}$},\emptyset,\raisebox{-.6ex}{$\begin{array}[t]{*{3}c}\\
\lr{9}\\
&\lr{8}\\
&\lr{6}&\lr{7}\\
\end{array}$}
}
>>> Tableaux.options.convention="russian"
>>> print(tex_from_array_tuple([[[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]],[],[[None,Integer(6),Integer(7)],[None,Integer(8)],[Integer(9)]]]))
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\cline{1-2}
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\\cline{1-3}
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\\cline{1-3}
\end{array}$}},\emptyset,\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\cline{1-1}
\lr{\rotatebox{-45}{9}}\\\cline{1-2}
&\lr{\rotatebox{-45}{8}}\\\cline{2-3}
&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\\cline{2-3}
\end{array}$}}
}
>>> print(tex_from_array_tuple([[[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(5)]],[],[[None,Integer(6),Integer(7)],[None,Integer(8)],[Integer(9)]]], with_lines=False))
{\def\lr#1{\multicolumn{1}{@{\hspace{.6ex}}c@{\hspace{.6ex}}}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\\
\lr{\rotatebox{-45}{4}}&\lr{\rotatebox{-45}{5}}\\
\lr{\rotatebox{-45}{1}}&\lr{\rotatebox{-45}{2}}&\lr{\rotatebox{-45}{3}}\\
\end{array}$}},\emptyset,\raisebox{-.6ex}{\rotatebox{45}{$\begin{array}[t]{*{3}c}\\
\lr{\rotatebox{-45}{9}}\\
&\lr{\rotatebox{-45}{8}}\\
&\lr{\rotatebox{-45}{6}}&\lr{\rotatebox{-45}{7}}\\
\end{array}$}}
}

>>> Tableaux.options._reset()

sage.combinat.output.tex_from_skew_array(array, with_lines=False, align='b')[source]¶

This function creates latex code for a “skew composition” array. That is, for a two dimensional array in which each row can begin with an arbitrary number None’s and the remaining entries could, in principle, be anything but probably should be strings or integers of similar width. A row consisting completely of None’s is allowed.

INPUT:

array – the array
with_lines – (default: False) if True lines are drawn, if False they are not
align – (default: 'b') determine the alignment on the latex array environments

EXAMPLES:

Sage

sage: array=[[None, 2,3,4],[None,None],[5,6,7,8]]
sage: print(sage.combinat.output.tex_from_skew_array(array))
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
&\lr{2}&\lr{3}&\lr{4}\\
&\\
\lr{5}&\lr{6}&\lr{7}&\lr{8}\\
\end{array}$}

Python

>>> from sage.all import *
>>> array=[[None, Integer(2),Integer(3),Integer(4)],[None,None],[Integer(5),Integer(6),Integer(7),Integer(8)]]
>>> print(sage.combinat.output.tex_from_skew_array(array))
\raisebox{-.6ex}{$\begin{array}[b]{*{4}c}\\
&\lr{2}&\lr{3}&\lr{4}\\
&\\
\lr{5}&\lr{6}&\lr{7}&\lr{8}\\
\end{array}$}