Montmort's problem, Burnside's Lemma and Bell's numbers

Abstract

For a set of n discernible objects, the probability of choosing at random a permutation with m objects fixed is determined with the help of the generating function method. It is shown that the expected number of objects left fixed is one, which represents a special interpretation of Burnside's Lemma. Furthermore, the higher moments about the origin are represented by Stirling's numbers of the second kind, or simpler, by Bell's numbers, and the factorial moments are all one. A short discussion of an estimation problem concludes the paper.

Darbe nagrinejami klasikiniai kombinatorikos uždaviniai su tam tikra tikimybine interpretacija. Autorius taiko generuojančiu funkciju metoda ivairiems momentams skaičiuoti. Kai kurios iš irodomu straipsnyje formuliu nera gerai žinomos kombina‐torineje analizeje. Kaip atskiri šiu formuliu rezultatai gaunami klasikiniai Stirlingo ir Belo skaičiu saryšiai. Straipsnyje pareikta trumpa nagrinejamu uždaviniu apžvalga.

Key words:

• Bell's numbers
• Burnside's lemma
• matching problem
• Stirling's numbers