partition
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partition [2019/03/07 09:31] 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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| }$$ | }$$ | ||
| This is a partition, where four upper and three lower points are divided into four blocks. One block consists of the upper point 1 and lower points 2 and 3. Second block consists of upper point 2 and the first lower point. Finally the last two blocks contain only one point, namely the third and the fourth point of the upper row. | This is a partition, where four upper and three lower points are divided into four blocks. One block consists of the upper point 1 and lower points 2 and 3. Second block consists of upper point 2 and the first lower point. Finally the last two blocks contain only one point, namely the third and the fourth point of the upper row. | ||
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| + | In this case, we again can define non-crossing partitions in the natural way -- a partition $p\in\Pscr(k,l)$ is non-crossing if and only if there is a graphical representation of $p$, where the nodes representing different blocks do not cross (following the rule that the nodes can be drawn only between the two lines). This corresponds to the original definition if we consider the so-called //cyclic order// of the points. That is, the point in the right top corner is the lowest, then follow all the points on the upper row from right to left, then comes the left bottom point and going on the bottom line from left to right we reach the greatest point in the bottom right corner. | ||
| ==== Word representation ==== | ==== Word representation ==== | ||
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| $$p=\mathsf{ccfbcfee}$$ | $$p=\mathsf{ccfbcfee}$$ | ||
| + | ===== Further reading ===== | ||
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| + | * [[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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