On distributions, partitions and derangements (with computer program)

• Main Authors: Jimenez, Maritess J., Gutierrez, Jeffrey John M.
• Format: text
• Language: English
• Published: Animo Repository 1993
• Subjects: Partitions (Mathematics); Set theory; Programming (Mathematics); Functions
• Online Access: https://animorepository.dlsu.edu.ph/etd_bachelors/16128
• Institution: De La Salle University

This thesis presents a detailed discussion of three important topics in combinatorial mathematics, namely, distributions, partitions and derangements. In the basic distribution model, there is a collection of objects that are distibuted to a group of possible recipients which are called boxes. The basic distribution model may take the following variations: (a) distribtution of distinct objects into identical boxes (b) distribution of identical objects into distinct boxes and (c) distribution of distinct objects into identical boxes. Since each object goes to exactly one box, the number of distributions of distinct objects into distinct boxes can be derived by counting the number of arbitrary functions from a set of objects to a set of boxes.Sometimes, the number of objects inside the boxes is specified. In the case where each box must hold at most one object, the number of distributions can be derived by counting the number of injections from a set of distinct objects to a set of distinct boxes. Likewise, the number of distributions of distinct objects into distinct boxes where each box holds at least one object can be derived by counting the number of surjections from a set of distinct objects to a set of distinct boxes.In deriving the number of distributions of identical objects into distinct boxes, and the number of distributions of distinct objects into identical boxes, an application of the Corespondence Principle is needed. Likewise, the Corespondence Principle is required in counting the total number of partitions of a given set. All of the formulas used for the derangement problem are results given by Hanson et al. in their article Matchings, Derangements, Rencontres. The rest of the formulas came from C. Chuan-Chong and K. Khee-Meng in their book entitled Principles and Techniques in Combinatorics and B.W. Jackson and D. Thoro in their Applied Combinatorics with Problem Solving. The researchers simplified the proofs and devised a computer program to solve the different problems presented in this problem as well as other similar problems.