ТНЕ ASYMPTOTIC OF THE BELL‘S NUMBERS SEQUENCE

Abstract

Bell‘s numbers B(s) defines the amount partitions of s-element set and with growth s they have an exponentiale growth. That‘s why the asymptotic‘s investigation s >> 1 of sequence {B(s)} of Bell‘s numbers B(s) becomes actual, for example, if do the following combinatorial sum. Let‘s take a discrete space of elementary event containing s points with given law of probability distribution p1; . . . ; ps, p1 + . . . + ps = 1. On configurations of partitions one should define such a partition at which minimum of informational Shanon‘s entropy is gained. One can face with this problem when the optimization of block-control of difficult cybernetic systems is present. In this work some asymptotic properties of sequence of Bell‘s numbers are considered. The main result of work represents the correlation: lims→∞ B(s)B(s + 2) B2(s + 1) = 1, where B(s); B(s + 1); B(s + 2) — Bell‘s numbers with numerals s; s + 1; s + 2. This result shows that asymptotical sequence of Bell‘s numbers behaved themselves geometrical progression with denominator x∗ = B(s + 1)/B(s). In the frames of additive presentation of Bell‘s numbers with the help of Stirling‘s numders the asymptotics is set up B(s) St(s; n∗) (n ∗ ) s/(n ∗ )!, where n∗ = [x∗]. Thus, a new class of sequences is up, the topology of which is characterized by the asymptotics in the form of the geometrical progression. Thus, a new class of sequences is established, the topology of wich is characterized by asymptotics in the form of geometrical progression.