Geodesic
Random partitions induced by random maps
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 24, Tome 454 (2016), pp. 195-215 Cet article a éte moissonné depuis la source Math-Net.Ru

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The lattice of the set partitions of $[n]$ ordered by refinement is studied. Given a map $\phi\colon[n]\to[n]$, by taking preimages of elements we construct a partition of $[n]$. Suppose $t$ partitions $p_1,p_2,\dots,p_t$ are chosen independently according to the uniform measure on the set of mappings $[n]\to[n]$. The probability that the coarsest refinement of all $p_i$'s is the finest partitions $\{\{1\},\dots,\{n\}\}$ is shown to approach $1$ for any $t\geq3$ and $e^{-1/2}$ for $t=2$. The probability that the finest coarsening of all $p_i$'s is the one-block partition is shown to approach $1$ if $t(n)-\log n\to\infty$ and $0$ if $t(n)-\log n\to-\infty$. The size of the maximal block of the finest coarsening of all $p_i$'s for a fixed $t$ is also studied. 
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}
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D. Krachun; Yu. Yakubovich. Random partitions induced by random maps. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 24, Tome 454 (2016), pp. 195-215. http://geodesic.mathdoc.fr/item/ZNSL_2016_454_a11/

[1] E. R. Canfield, “Meet and join within the lattice of set partitions”, Electron. J. Combin., 8:1 (2001), Research Paper 15, 8 pp. | MR | Zbl

[2] W. Y. C. Chen, D. G. L. Wang, “Minimally intersecting set partitions of type $B$”, Electron. J. Combin., 17:1 (2010), Research Paper 22, 16 pp. | MR

[3] J. Engbers, A. Hammett, “Trivial meet and join within the lattice of monotone triangles”, Electron. J. Combin., 21:3 (2014), Paper 3.13, 15 pp. | MR | Zbl

[4] P. Flajolet, P. J. Grabner, P. Kirschenhofer, H. Prodinger, “On Ramanujan's $Q$-function”, J. Comput. Appl. Math., 58:1 (1995), 103–116 | DOI | MR | Zbl

[5] L. C. Hsu, “Note on an asymptotic expansion of the $n$th difference of zero”, Ann. Math. Statist., 19 (1948), 273–277 | DOI | MR | Zbl

[6] S. Janson, T. Łuczak, A. Rucinski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000 | MR | Zbl

[7] O. Johnson, C. Goldschmidt, “Preserving of log-concavity on summation”, ESAIM: Probab. Statist., 10 (2006), 206–215 | DOI | MR | Zbl

[8] B. Pittel, “Where the typical set partitions meet and join”, Electron. J. Combin., 7 (2000), Research Paper 5, 15 pp. | MR | Zbl

[9] D. Knut, Iskusstvo programmirovaniya, v. 1, Osnovnye algoritmy, Vilyams, M., 2015 | MR

[10] R. Stenli, Perechislitelnaya kombinatorika, Mir, M., 1990 | MR

[11] E. G. Tsylova, “Veroyatnostnye metody polucheniya asimptoticheskikh formul dlya obobschennykh chisel Stirlinga”, Statist. sbornik, 8, 1991, 165–178 | MR