.. _library_partitions:

partitions

This library provides predicates for generating and querying set partitions of lists. A set partition divides a list into non-empty, non-overlapping blocks whose union is the original list. The following categories of predicates are provided:

• Generation operations - Predicates for generating all set partitions and set partitions with an exact number of blocks.
• Ordering variants - Predicates that support an additional order argument (default or lexicographic) for controlling output order.
• Distinct-value generation - Predicates for generating set partitions while deduplicating equal-valued partitions after canonicalizing block order.
• Indexed access - Predicates for retrieving partitions and distinct partitions by zero-based index and for recovering the index of a partition.
• Random selection - Predicates for selecting random partitions and taking random samples, with distinct-value and exact-block-count variants.
• Lexicographic stepping - Predicates for moving to the next or previous distinct partition value in lexicographic order.
• Counting operations - Predicates for counting set partitions and distinct set partitions, including exact-block counts.

  Base predicates are position-sensitive, matching the existing combinatorics libraries. Inputs with repeated values can therefore yield duplicate-valued partitions. Distinct predicates collapse those equal-valued results.

  The lexicographic stepping predicates operate on the distinct partition view, using the same canonical block ordering as the distinct generation predicates.

  The counting predicates are connected to Bell numbers (for all partitions of an N-element list) and Stirling numbers of the second kind (for partitions with an exact number of blocks).

  Dedicated arrangements, cartesian_products, combinations, derangements, multisets, permutations, and subsequences libraries are also available for focused APIs on related operations.

API documentation

Open the `../../apis/library_index.html#partitions <../../apis/library_index.html#partitions>`__ link in a web browser.

Loading

To load all entities in this library, load the loader.lgt file:

::

| ?- logtalk_load(partitions(loader)).

Testing

To test this library predicates, load the tester.lgt file:

::

| ?- logtalk_load(partitions(tester)).