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')#

Return an ascii art table of data.

EXAMPLES:

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 │
    └───┘
sage.combinat.output.ascii_art_table_russian(data, use_unicode=False, compact=False)#

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

EXAMPLES:

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 ╱
  ╲ ╱
sage.combinat.output.box_exists(tab, i, j)#

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 – a list of lists

  • i – first coordinate

  • j – second coordinate

sage.combinat.output.tex_from_array(array, with_lines=True)#

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

INPUT:

  • array – a list of list

  • with_lines – a 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: 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()
sage.combinat.output.tex_from_array_tuple(a_tuple, with_lines=True)#

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

INPUT:

  • a_tuple – a tuple of lists of lists

  • with_lines – a 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: 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()
sage.combinat.output.tex_from_skew_array(array, with_lines=False, align='b')#

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') Determines the alignment on the latex array environments

EXAMPLES:

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