The paper discusses the specific features and presents the procedure and the results of studying a wide variety of specific combinatorial schemes in the pre-asymptotic region of their parameters change. It is suggested that the schemes are analyzed by an unconventional quality analysis of their outcomes, the result of which include quantitative characteristics.
Transpositions of lower lines of random permutations of size n with forbidden subsequences of size s<n are considered. The number of such transpositions are finded, their enumerations are reali-zed, the numeration problem of outcomes of the scheme is decided and their modeling is considered.
The following investigations are carried out in the general scheme of allocation of indistinguishable particles to indistinguishable cells and in the particular scheme with no empty cell allowed. A recurrence relation is found for the total number of outcomes of the particular scheme and an explicit expression for it is obtained. A relation between the numbers of outcomes of the general and particular schemes is found. A random process of successive allocation of a single particle to cells which provides us with an algorithm for solving the combinatorial problem and ﬁnding all the outcomes of allocation of a ﬁxed number of particles to cells in the scheme under consideration and ﬁnding the distribution of their probabilities is escribed. Various methods to simulate the states of the scheme and approximate the numberof its outcomes by means of stochastic simulation techniques are suggested.
The problems of a direct enumeration of the outcomes of scheme, finding their numbers, problems of their numbering and modelling are solved.
The chip of domino as the scheme of the random filling of a chip of general domino is determined with r ends and n number from 0 to (n − 1) on the ends of chips of all possible compositions with a repetition without the account of their order. The research of this scheme is conducted and analogical with the fixed minimum number r > m the outcome of random procedure of enumeration of the numbered outcomes of chip determination of their number, decisions from their problem of numeration (i.e. establishments of one-to-one accordance between numbers and types of the outcomes of the scheme), finding of their probabilistic distribution and modelling of their possible outcomes.
We consider several procedures to number all outcomes of a permutation scheme, establish a one-to-one correspondence between the outcome and its number generated in the numbering procedure, and give some methods to simulate the outcomes.
The scheme of allocating r distinguishable particles into n indistinguishable cells with k non-empty cells is studied along the directions of enumerative combinatorial analysis. They are the direct enumeration of the outcomes of the scheme with a particular discipline of their numbering, finding their numbers, establishing a one-to-one correspondence of the types of outcomes with their numbers, aka numbering problem in direct and inverse statements, and modeling of outcomes of the scheme.
We introduce a new characteristic of the outcome of a combination scheme, i. e. its steps, defined as differences between the neighbor-ring elements of the outcome arranged in the order of increase. We consider the procedure of enumerating all outcomes of a combina-tion scheme with a given restriction, establish the one-to-one cor-respondence between the outcomes and their numbers generated in the enumeration procedure, and give some methods to simulate the possible outcomes of the scheme.
A model of system maintenance which ensures secure operation of the system is investigated. To this end, mathematical model of a controlled semi-Markov process with catastrophes is used. The model allows to take into account the features of self-manifestation of failure (the time has an arbitrary distribution) and features of disaster occurrence (the time required to inflict irreparable harm and the possibility of irreparable harm not only at the first penetration into the system). Connection between reliability characteristics (failure-free operation and maintainability) and security features is established. The problem of optimization of the frequency of restoration work, in which the expectation for time until disaster is maximal, is solved.
In this article the model of controlled semi-Markov process with accidents with reference to a safety problem is investigated. Characteristics (indicators) of safety are entered. The mathematical model is used for the analysis of characteristics of safety of technical system which provides protection of object (information, the territory, etc.). Connection of characteristics of reliability (non-failure operation and maintainability) and safety characteristics is established. The situation of a choice of optimum strategy of management in the conditions of incomplete information on characteristics of reliability of system is analyzed.
The numerical method of calculation of inversions in permutations uses the method of graphs. As the result we calculate the recurrence for calculation of the numbers of inversions with the use of the bunchs of graphs and finding the probability distribution of the number of inversions with the increes of the size of the permutationThe scheme definded in its name is investigated by asi-ding some outcomes of the similar scheme without restriction. The number of outcomes of the scheme is obtained, the direct enumera-tion is fulfilled, their probability distribution is finded the problem of their numeration is solved and the scheme suggested.