Overview of the Iterator Package
The Iterator package exports constructors of efficient iterators over discrete structures.
Each iterator is an object with a ModuleIterator method. It can be used in for loops and in seq, add, or mul commands.
To reduce memory usage the iterators use a mutable data structure, a one-dimensional Array, as the output.
All constructors provide a compile option that is true by default. When true, the returned iterator is compiled.
Because these iterators use the state of the object, using the same iterator in independent loops is not allowed; an error is raised if that occurs. To loop over the the same discrete structure in independent loops, copy the object (use Object).
See Iterator/Details for details of an iterator object.
The following commands are available.
The following subpackages are available.
procedures for converting between permutations and inversion tables
procedures for operating on mixed-radix tuples
procedure for converting between tree representations
generate bounded compositions that sum to a constant
generate all (s,t)-combinations of zeros and ones in near-perfect order
generate t-combinations of a set
generate feasible ways to fill a rucksack
generate combinations in the lexicographic revolving-door Gray code
generate compositions of an integer
generate weighted compositions that sum to a constant
generate mixed-radix tuples
generate multiple sequences
subpackage for converting between permutations and inversion tables
subpackage for converting between tree representations
generate n-bit binary Gray code
generate a mixed-radix Gray code
generate Lyndon words that form a de Bruijn sequence
generate Lyndon words
generate positions of left-parentheses in pairs of nested parentheses
generate pairs of nested parentheses
generate partitions of a multiset
generate all partitions of an integer
generate fixed-size partitions of an integer
generate partitions of an integer in part-count form
generate permutations of a list
generate permutations with restrictions
generate a Cartesian product of lists and sets
create the product of iterators
generate fixed-size partitions of a set
generate set partitions with restricted growth strings in Gray code order
generate set partitions with restricted growth strings
compute the starting ranks and iterations suitable for parallelizing an iterator
generate binary trees of a given size
generate oriented forests of a given size
The Twelve-Fold Way
Richard Stanley, in Enumerative Combinatorics, categorizes common combinatorial selections using the cardinality of unrestricted, injective, and surjective functions between discrete domains in a 4×3 tableau called "The Twelve-fold Way." The following table reproduces this categorization, using the enumeration of ways to place balls into urns, and links to the appropriate iterator.
balls per urn
at most 1
at least 1
n labeled balls, m labeled urns
n unlabeled balls, m labeled urns
n labeled balls, m unlabeled urns
n unlabeled balls, m unlabeled urns
 Partitions of n distinct objects into m ordered parts.
 If n≤m there is one arrangement that satisfies this, otherwise none.
Index of interesting examples. The link goes to the help page; look in its Examples section for the example.
solve an alphametic puzzle (cryptarithm)
compute number of contingency tables
Dudney's century puzzle
solve Dudney's century puzzle
compute matrix permanent with Ryser's algorithm
compute number of distinct ranks of a poker hand
Simplified Dudney's century puzzle
solve simplified Dudney's century puzzle
Split list of floats
split a list of floating-point numbers into nearly equal sublists. Demonstrates the creation and use of parallelized iterators.
create a Young rectangle (specialization of a Young tableau)
Use Permute to construct an iterator over all permutations of the list [1,2,2,3].
P ≔ Permute⁡1,2,2,3:
Use a for-loop to iterate over the permutations.
1 2 2 3
1 2 3 2
1 3 2 2
2 1 2 3
2 1 3 2
2 2 1 3
2 2 3 1
2 3 1 2
2 3 2 1
3 1 2 2
3 2 1 2
3 2 2 1
The same output is more conveniently generated with the Print method. Here the number of iterations is limited and showrank option is used to display the rank.
1: 1 2 2 3
2: 1 2 3 2
3: 1 3 2 2
4: 2 1 2 3
5: 2 1 3 2
6: 2 2 1 3
7: 2 2 3 1
8: 2 3 1 2
9: 2 3 2 1
10: 3 1 2 2
Use a seq command to create the entire sequence.
Note the use of the square brackets, , to instantiate the Vector that is assigned to p. Without them, all values in the final expression sequence equal the last value because the p' evaluates to the Vector rather than its content. Here is what happens when the square brackets are omitted.
Using hasNext and getNext
Use Combination to generate all triplets of the integers 0 to 4. Extract the two procedures, hasNext and getNext, from the ModuleIterator method of the iterator object and use them in a while-loop.
M ≔ Combination⁡5,3:
hasNext,getNext ≔ ModuleIterator⁡M:
Construct an iterator over the 2-permutations of the list 1,1,2, use Object to create an identical, but independent, second iterator, and use both iterators in a dual-loop.
P ≔ Permute⁡1,1,2,2:
Q ≔ Object⁡P:
forpinPdoforqinQdoprint⁡p,qend doend do:
The Iterator package was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
Download Help Document