多角度处理分装模型的计数与概率计算

Multiangular Analysis of the Counting and Probability in the Occupancy Problem

从多个角度利用多种方法计算一类分装模型的计数,同时给出了相应的概率计算.分装模型就是将n个球分装到m个盒子中计数的模型.分装模型涉及到排列与组合、反演公式、容斥原理、Stirling数、生成函数及整数的分拆等组合数学中的大部分的计数方法.本文从组合数学的不同计数方法入手,详细叙述分装模型在不同情形下的解,深入剖析不同情形下解不同的原因.

The counting problems in the occupancy model are discussed in many methods from different points of view.The probabilities are also simultaneously given.The occupancy model is the problem of distributing n balls to m boxes.Most of the counting methods in combinatorics are used in the occupancy model,such as permutation,combination,inversion formula,inclusion-exclusion principle,stirling number,generating function and partition number.We elaborate the solutions of the occupancy problem in the different cases using many different methods.Finally,we analyze the reasons why the solutions are different.