Compositions of random functions on a finite set

Abstract

If we compose sufficiently many random functions on a finite set, then the composite function will be constant. We determine the number of compositions that are needed, on average. Choose random functions f1, f 2, f3, . . . independently and uniformly from among the nn functions from [n] into [n]. For t > 1, let gt = ft ○ ft-1 ○ ⋯ ○ f1 be the composition of the first t functions. Let T be the smallest t for which g t is constant(i.e. gt(i) = gt(j) for all i, j). We prove that E(T) ∼ 2n as n → ∞, where E(T) denotes the expected value of T.