Geodesic
Where the typical set partitions meet and join
The electronic journal of combinatorics, Tome 7 (2000)
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The lattice of the set partitions of $[n]$ ordered by refinement is studied. Suppose $r$ partitions $p_1,\dots,p_r$ are chosen independently and uniformly at random. The probability that the coarsest refinement of all $p_i$'s is the finest partition $\bigl\{\{1\},\dots,\{n\}\bigr\}$ is shown to approach $0$ for $r=2$, and $1$ for $r\ge 3$. The probability that the finest coarsening of all $p_i$'s is the one-block partition is shown to approach $1$ for every $r\ge 2$.
DOI : 10.37236/1483
Classification : 05A18, 05A19, 05C80, 60C05, 06A07
Mots-clés : set partition lattice, join operations, enumeration, random, limiting probabilities, random partitions 
@article{10_37236_1483,
     author = {Boris Pittel},
     title = {Where the typical set partitions meet and join},
     journal = {The electronic journal of combinatorics},
     year = {2000},
     volume = {7},
     doi = {10.37236/1483},
     zbl = {0940.05007},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1483/}
}
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Boris Pittel. Where the typical set partitions meet and join. The electronic journal of combinatorics, Tome 7 (2000). doi: 10.37236/1483

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