Geodesic
Partitions without small blocks and $r$-associated Bell polynomials in a parametric model: probabilistic-statistical analysis
Matematičeskie voprosy kriptografii, Tome 10 (2019) no. 1, pp. 27-40 
G. I. Ivchenko; Yu. I. Medvedev. Partitions without small blocks and $r$-associated Bell polynomials in a parametric model: probabilistic-statistical analysis. Matematičeskie voprosy kriptografii, Tome 10 (2019) no. 1, pp. 27-40. http://geodesic.mathdoc.fr/item/MVK_2019_10_1_a2/
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     title = {Partitions without small blocks and $r$-associated {Bell} polynomials in a parametric model: probabilistic-statistical analysis},
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Voir la notice de l'article provenant de la source Math-Net.Ru

On the set of all partitions of an $n$-element set $X_n = \{1, 2,\dots, n\}$ into blocks with sizes exceeding the number $r\geqslant 0$ a probability measure is defined such that for each partition with $k$ blocks its probability is proportional to $\theta^k$, where $\theta>0$ is the parameter of the measure. The asymptotic normality of the number of blocks in a random partition of $X_n$ in this model is proved, a statistical test for the uniformity hypothesis $H_0 :\, \theta = 1$ against the alternatives $H_1 :\, \theta \ne 1$ is constructed.

[1] Sachkov V. N., Vvedenie v kombinatornye metody diskretnoi matematiki, 2-e izd., MTsNMO, M., 2004, 424 pp.

[2] Bell E. T., “Exponential polynomials”, Ann. Math., 35 (1934), 258–277 | DOI | MR

[3] Ivchenko G. I., Medvedev Yu. I., “Sluchainye kombinatornye ob'ekty”, Doklady RAN, 396:2 (2004), 151–154 | Zbl

[4] Charalambides Ch., “The generalized Stirling and C-numbers”, Sankhya (Ser. A), 36 (1974), 419–436 | MR | Zbl

[5] de Brein N. G., Asimptoticheskie metody v analize, IL, M., 1961, 248 pp.

[6] Harper L., “Stirling behavior is asymptotically normal”, Ann. Math. Stat., 38:2 (1961), 410–414 | DOI | MR

[7] Hwang H. K., “On convergence rates in the central limit theorems for combinatorial structures”, European J. Combin., 19:3 (1998), 329–343 | DOI | MR | Zbl

[8] Riordan Dzh., Vvedenie v kombinatornyi analiz, IL, M., 1963, 288 pp.

[9] Moll V. H., Ramirez J. L., Villamizar D., Combinatorial and arithmetical properties of the restricted and associated Bell and factorial numbers, 2017, arXiv: 1707.08138v2 | MR