Probability Distributions Related to Random Mappings

Abstract

A Random Mapping Space $(X, \mathcal{J}, P)$ is a triplet, where $X$ is a finite set of elements $x$ of cardinality $n, \mathcal{J}$ is a set of transformations $T$ of $X$ into $X$, and $P$ is a probability measure over $\mathcal{J}$. In this paper, four choices of $\mathcal{J}$ are considered (I) $\mathcal{J}$ is the set of all transformations of $X$ into $X$. (II) $\mathcal{J}$ is the set of all transformations of $X$ into $X$ such that for each $x \varepsilon X Tx \neq x$. (III) $\mathcal{J}$ is the set of one-to-one mappings of $X$ onto $X$. (IV) $\mathcal{J}$ is the set of one-to-one mappings of $X$ onto $X$, such that for each $x \varepsilon X, Tx \neq x$. In each case $P$ is taken as the uniform probability distribution over $\mathcal{J}$. If $x \varepsilon X$ and $T \varepsilon \mathcal{J}$, we will define $T^kx$ as the $k$th iteration of $T$ on $x$, where $k$ is an integer, i.e. $T^kx = T(T^{k-1}x)$, and $T^0x = x$ for all $x$. The reader should note that, in general, $T^kx, k < 0$, may not exist or may not be uniquely determined. If for some $k \geqq 0, T^kx = y$, then $y$ is said to be a $k$th image of $x$ in $T$. The set of successors of $x$ in $T, S_T(x)$ is the set of all images of $x$ in $T$, i.e., $S_T(x) = \{x, Tx, T^2x, \cdots, T^{n-1}x\},$ which need not be all distinct elements. If for some $k \leqq 0, T^kx = y, y$ is said to be a $k$th inverse of $x$ in $T$. The set of all $k$th inverses of $x$ in $T$ is $T^{(k)} (x)$ and $P_T(x) = \mathbf{\bigcup}^0_{k=-n} T^{(k)} (x)$ is the set of predecessors of $x$ in $T$. If there exists an $m > 0$, such that $T^mx = x$, then $x$ is a cyclical element of $T$ and the set of elements $x, Tx, T^2x, \cdots, T^{m-1}x$ is the cycle containing $x, C_T(x)$. If $m$ is the smallest positive integer for which $T^m x = x$, then $C_T(x)$ has cardinality $m$. We note further an interesting equivalence relation induced by $T$. If there exists a pair of integers $k_1, k_2$ such that $T^{k_1}x = T^{k_2}y,$ then $x \sim y$ under $T$. It is readily seen that this is in fact an equivalence, and hence decomposes $X$ into equivalence classes, which we shall call the components of $X$ in $T$; and designate by $K_T(x)$ the component containing $x$. We define $s_T(x)$ to be the number of elements in $S_T(x), p_T(x)$ to be the number of elements in $P_T(x)$, and $l_T(x)$ to be the number of elements in the cycle contained in $K_T(x)$ (i.e. $l(x) =$ the number of elements in $C_T(x)$ if $x$ is cyclical). We designate by $q_T$ the number of elements of $X$ cyclical in $T$, and by $r_T$ the number of components of $X$ in $T$. Rubin and Sitgreaves [9] in a Stanford Technical Report have obtained the distributions of $s, p, l, q,$ and have given a generating function for the distribution of $r$ in case I. Folkert [3], in an unpublished doctoral dissertation has obtained the distribution of $r$ in cases I and II. The distribution of $r$ in case III is classical and may be found in Feller [2], Gontcharoff [4], and Riordan [8]. In the present paper, a number of these earlier results are rederived and extended. Specifically, for cases I and II, we compute the probability distributions of $s, p, l, q$ and $r$. In cases III and IV the distributions of $l$ and $r$ are given. In addition some asymptotic distributions and low order moments are obtained. For the convenience of the reader, an index of notations having a fixed meaning is provided in the appendix to the paper.