Н.Ю. Энатская, Е.Р. Хакимуллин, А.В. Колчин.
Анализ схемы размещения неразличимых частиц по неразличимым ячейкам
// Труды КарНЦ РАН. No 4. Сер. Математическое моделирование и информационные технологии. 2014. C. 143-154
N.Yu. Enatskaya, Е.R. Khakimullin, А.V. Kolchin. Analysis of a scheme of allocating indistinguishable particles to indistinguishable cells // Transactions of Karelian Research Centre of Russian Academy of Science. No 4. Mathematical Modeling and Information Technologies. 2014. Pp. 143-154
Keywords: enumerative combinatorics, allocation scheme
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 described. Various methods to simulate the states of the scheme and approximate the number of its outcomes by means of stochastic simulation techniques are suggested.