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.