## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 26, Number 3 (1955), 512-517.

### Probability of Indecomposability of a Random Mapping Function

#### Abstract

Consider a finite set $\Omega$ of $N$ points and a single-valued function $f(x)$ on $\Omega$ into $\Omega.$ In case the mapping is one-to-one, it is a permutation of the points of $\Omega;$ we shall be concerned with more general mappings. Any mapping function effects a decomposition of the set into disjoint, minimal, non-null invariant subsets, as $\Omega = \omega_1 + \omega_2 + \cdots + \omega_k,$ where $f(\omega_i) \subset \omega_i$ and $f^{-1}(\omega_i) \subset \omega_i.$These subsets have been referred to as trees and as components of the mapping; we shall say that $f,$ as above, decomposes the set into $k$ components. Metropolis and Ulam [1] defined a random mapping by a uniform probability distribution over the $\Omega^\Omega$ sample points of $f(x)$ and posed the problem of finding the expected number of components. Kruskal [2] subsequently solved this problem. In this paper, we consider a related problem, namely, what is the probability that a random mapping is indecomposable, i.e., that the minimal non-null set $\omega$ for which $f(\omega) = \omega$ and $f^{-1}(\omega) = \omega,$ is the whole set $\omega = \Omega?$ This problem is solved in general, as is, also, an analogous problem for a specialized random mapping of some interest in social psychology. Finally, we examine the asymptotic behavior of these probabilities.

#### Article information

**Source**

Ann. Math. Statist., Volume 26, Number 3 (1955), 512-517.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177728496

**Digital Object Identifier**

doi:10.1214/aoms/1177728496

**Mathematical Reviews number (MathSciNet)**

MR70869

**Zentralblatt MATH identifier**

0065.11604

**JSTOR**

links.jstor.org

#### Citation

Katz, Leo. Probability of Indecomposability of a Random Mapping Function. Ann. Math. Statist. 26 (1955), no. 3, 512--517. doi:10.1214/aoms/1177728496. https://projecteuclid.org/euclid.aoms/1177728496