Young tableaux and other mutually describing sequences

We introduce a transformation on integer sequences for which the set of images is in bijective correspondence with the set of Young tableaux. We use this correspondence to show that the set of images, known as ballot sequences, is also the set of double points of our transformation. In the second part, we introduce other transformations of integer sequences and show that, starting from any sequence, repeated applications of the transformations eventually produce a fixed point (a self-describing sequence) or a double point (a pair of mutually describing sequences). Counting equal terms Let A be the set of finite integer sequences a = a1a2 . . . with 1 ≤ ai ≤ i, for all indices. Define a transformation of sequences β : A → A by β(a)i = #{j | j ≤ i, aj = ai}. Thus β(a)i counts the number of terms in the sequence a that are equal to ai and appear in the initial segment of a consisting of the first i positions. Therefore, in some sense, the sequence β(a) describes the sequence a. It is easy to see that the only fixed point of β is the sequence 1. However, there are many double points, i.e., sequences a for which β(a) = a. If a is a double point so is b = β(a), we have a = β(b), and the sequences a and b mutually describe each other. Theorem 1. The set of double points of β of length n in A