On perfect, amicable, and sociable chains

Let x = (x0, . . . , xn−1) be an n-chain, i.e., an n-tuple of non-negative integers < n. Consider the operator s : x 7→ x′ = (x0, . . . , x ′ n−1), where x ′ j represents the number of j’s appearing among the components of x. An n-chain x is said to be perfect if s(x) = x. For example, (2,1,2,0,0) is a perfect 5-chain. Analogously to the theory of perfect, amicable, and sociable numbers, one can define from the operator s the concepts of amicable pair and sociable group of chains. In this paper we give an exhaustive list of all the perfect, amicable, and sociable chains.