compile = truefalse

True means compile the iterator. The default is true.

rank = nonnegint

Specify the starting rank of the iterator. The default is one. Passing a value greater than one causes the iterator to skip the lower ranks; this can be useful when parallelizing iterators. The starting rank reverts to one when the iterator is reset, reused, or copied.

The NestedParentheses command returns an iterator that generates all nested n-pairs of parentheses.

A left parenthesis is represented by 0, a right parenthesis by 1.

The n parameter specifies the number of pairs of parentheses in the output.

See Iterator[Trees] for procedures to convert this format (A) to other formats for nested parentheses and binary trees.

$\mathrm{with}\left(\mathrm{Iterator}\right)\:$

$M≔\mathrm{NestedParentheses}\left(4\right)\:$

$\mathrm{Print}\left(M\,'\mathrm{showrank}'\right)\:$

 1: 0 1 0 1 0 1 0 1
 2: 0 1 0 1 0 0 1 1
 3: 0 1 0 0 1 1 0 1
 4: 0 1 0 0 1 0 1 1
 5: 0 1 0 0 0 1 1 1
 6: 0 0 1 1 0 1 0 1
 7: 0 0 1 1 0 0 1 1
 8: 0 0 1 0 1 1 0 1
 9: 0 0 1 0 1 0 1 1
10: 0 0 1 0 0 1 1 1
11: 0 0 0 1 1 1 0 1
12: 0 0 0 1 1 0 1 1
13: 0 0 0 1 0 1 1 1
14: 0 0 0 0 1 1 1 1

$\mathrm{Number}\left(M\right)$

$14$

$\mathrm{Unrank}\left(M\,4\right)$

$\left[\begin{array}{cccccccc}0& 1& 0& 0& 1& 0& 1& 1\end{array}\right]$

$N≔\mathrm{Object}\left(M\,\mathrm{rank}=4\right)\:$

$\mathrm{seq}\left(v\left[\right]\,v=N\right)$

$\left[\begin{array}{cccccccc}0& 1& 0& 0& 1& 0& 1& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 1& 0& 0& 0& 1& 1& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 1& 0& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 1& 0& 1& 1& 0& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 1& 0& 1& 0& 1& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 1& 0& 0& 1& 1& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 0& 1& 1& 1& 0& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 0& 1& 1& 0& 1& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 0& 1& 0& 1& 1& 1\end{array}\right],\left[\begin{array}{cccccccc}0& 0& 0& 0& 1& 1& 1& 1\end{array}\right]$

$\mathrm{str1}≔\mathrm{String}\left(\mathrm{NestedParentheses}\left(15\,\mathrm{rank}={10}^6\right)\right)$

$\mathrm{str1}≔''NestedParentheses(15,\text{'}rank\text{'}\; =\; 1000000)''$

$\mathrm{iter}≔\mathrm{NestedParentheses}\left(15\right)\:$

$\mathrm{cnt}≔1\:$

$$\begin{gathered} \mathbf{for}V\mathbf{in}\mathrm{iter}\mathbf{while}\mathrm{cnt}<{10}^6-1\mathbf{do} \\ \mathrm{cnt}≔\mathrm{cnt}+1 \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

$\mathrm{str2}≔\mathrm{String}\left(\mathrm{iter}\right)$

$\mathrm{str2}≔''NestedParentheses(15)''$

$\mathrm{evalb}\left(\mathrm{str1}=\mathrm{str2}\right)$

$\mathrm{false}$

Knuth, Donald Ervin. The Art of Computer Programming, volume 4, fascicle 4; generating all trees, sec. 7.2.1.6, generating all trees, algorithm P, nested parentheses in lexicographic order, p. 5.

ibid, Algorithm U, unrank a string of nested parentheses, p. 14.

The Iterator[NestedParentheses] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.