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Instant-use add-on functions for the Wolfram Language
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SetPartitions
Give all possible ways to partition a set into blocks, ignoring the order of blocks and order within blocks
Contributed by:
Wolfram Staff
ResourceFunction["SetPartitions"][set] returns the list of set partitions of set. | |
ResourceFunction["SetPartitions"][n] returns the list of set partitions of {1,2,…, n}. |
Details and Options
A set partition of a set S is an unordered set of non-empty disjoint subsets of S (called blocks) whose union is S.
Both the order of the blocks and the order within each block are ignored.
Examples
Basic Examples (2)
There are five set partitions of a three-element set:
| In[1]:= | ![sp = ResourceFunction["SetPartitions"][{a, b, c}]](https://www.wolframcloud.com/obj/resourcesystem/images/23c/23cfd32b-8215-488e-875f-44585e164f6a/091908ef6023810f.png){kind=small} |
| Out[1]= |  |
The number of set partitions of a set with n elements is given by the nth Bell number:
| In[2]:= |  |
| Out[2]= |  |
| In[3]:= | ![BellB[3]](https://www.wolframcloud.com/obj/resourcesystem/images/23c/23cfd32b-8215-488e-875f-44585e164f6a/495e08a9af456d42.png){kind=small} |
| Out[3]= |  |
Scope (1)
Here is a compact way to see the blocks:
| In[4]:= | ![Map[Row, ResourceFunction["SetPartitions"][5], {2}]](https://www.wolframcloud.com/obj/resourcesystem/images/23c/23cfd32b-8215-488e-875f-44585e164f6a/795752853a411e55.png){kind=small} |
| Out[4]= |  |
Publisher
Related Links
Requirements
Wolfram Language 11.3 (March 2018) or above
Version History
- 2.0.0 – 13 March 2019
- 1.0.0 – 11 March 2019
Related Resources
Related Symbols
License Information
This work is licensed under a Creative Commons Attribution 4.0 International License