The distribution of the number of fixed points corresponding to elements of a~symmetric semigroup with the condition $\sigma^{h+1}=\sigma^h$, and the number of trees with the altitudes less or equal to~$h$
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 676-687
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A one-to-one correspondence is set between elements $\sigma$ of the symmetric semigroup $\sigma_n$ with the condition $\sigma^{h+1}=\sigma^h$ and graphs $\Gamma_h$ consisting of root trees with the altitudes less or equal to $h$. The number of fixed points of elements $\sigma\in\sigma_n^h$ chosen at random and the number of components of the graphs $\Gamma_h$ are shown to be asymptotically normal as $n\to\infty$. When no restriction is laid on the altitude of trees, the number of components in corresponding graphs is proved to be asymptotically (as $n\to\infty$) distributed according to Poisson law. Asymptotic formulas are derived for the number of root trees with enumerated vertices with the altitudes less or equal to $h$ and for the number of graphs composed of such trees.
@article{TVP_1971_16_4_a5,
author = {V. N. Sa\v{c}kov},
title = {The distribution of the number of fixed points corresponding to elements of a~symmetric semigroup with the condition $\sigma^{h+1}=\sigma^h$, and the number of trees with the altitudes less or equal to~$h$},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {676--687},
publisher = {mathdoc},
volume = {16},
number = {4},
year = {1971},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a5/}
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AU - V. N. Sačkov
TI - The distribution of the number of fixed points corresponding to elements of a~symmetric semigroup with the condition $\sigma^{h+1}=\sigma^h$, and the number of trees with the altitudes less or equal to~$h$
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1971
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VL - 16
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V. N. Sačkov. The distribution of the number of fixed points corresponding to elements of a~symmetric semigroup with the condition $\sigma^{h+1}=\sigma^h$, and the number of trees with the altitudes less or equal to~$h$. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 676-687. http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a5/