partition
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partition [2019/03/07 09:50] d.gromada |
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| ==== Graphical representation on one line ==== | ==== Graphical representation on one line ==== | ||
| - | We can represent elements $p\in\Part(k)$, that is, partitions of a set $\{1,\dots,k\}$ as diagrams as follows. Put $k$ points on one line representing the elements of the set $\{1,\dots,k\}$. For each block $V\in p$ connect all the points representing elements of $V$ by a node. | + | We can represent elements $p\in\Pscr(k)$, that is, partitions of a set $\{1,\dots,k\}$ as diagrams as follows. Put $k$ points on one line representing the elements of the set $\{1,\dots,k\}$. For each block $V\in p$ connect all the points representing elements of $V$ by a node. |
| For example, consider $k=8$ and a partition $p=\{\{1,2,5\},\{3,6\},\{4\},\{7,8\}\}$. This can be represented by the following diagram | For example, consider $k=8$ and a partition $p=\{\{1,2,5\},\{3,6\},\{4\},\{7,8\}\}$. This can be represented by the following diagram | ||
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| * [[wp>Partition_of_a_set|Partition of a set]] | * [[wp>Partition_of_a_set|Partition of a set]] | ||
| + | * Richard P. Stanley. //Enumerative Combinatorics Vol. 1.// [[https://www.cambridge.org/us/academic/subjects/mathematics/discrete-mathematics-information-theory-and-coding/enumerative-combinatorics-volume-1-2nd-edition?format=PB&isbn=9781107602625|Cambridge University Press, 2011]] [[http://www-math.mit.edu/~rstan/ec/ec1.pdf|Manuscript available on-line.]] | ||
| + | * Alexandru Nica and Roland Speicher. //Lectures on the Combinatorics of Free Probability.// [[https://www.cambridge.org/core/books/lectures-on-the-combinatorics-of-free-probability/D8FC5F2DCCF6C34BB3164E0C0C29437F|Cambridge University Press, 2006.]] [[https://www.math.uni-sb.de/ag/speicher/publikationen/Nica-Speicher.pdf|Manuscript available on-line.]] | ||
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