Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: elf-describing sequences and the Catalan family tree | Self-describing sequences and the Catalan family tree Zoran Sunik Department of Mathematics Texas A M University College Station TX 77843-3368 USA Submitted Mar 19 2002 Accepted May 20 2003 Published May 29 2003 MR Subject Classifications 05A15 05C05 11Y55 Abstract We introduce a transformation of finite integer sequences show that every sequence eventually stabilizes under this transformation and that the number of fixed points is counted by the Catalan numbers. The sequences that are fixed are precisely those that describe themselves every term t is equal to the number of previous terms that are smaller than t. In addition we provide an easy way to enumerate all these self-describing sequences by organizing them in a Catalan tree with a specific labelling system. Prefix ordered sequences and rooted labelled trees The following connection between prefix ordered sequences and rooted labelled trees is well known and we briefly mention only the instance which is useful for our considerations. Let A be the set of finite integer sequences a a0 a1 . with the property that 0 ai i for all indices. We order the sequences in A by the prefix relation . ao ai . an bo bi . bm if n m and ai bi for i 0 . n. The sequences in A can be organized in a rooted labelled tree T which reflects the prefix order relation. The root of the tree T is labelled by 0. Every vertex that is at distance n from the root has n 2 children labelled by 0 1 . n n 1 see Figure 1 . The vertices whose distance to the root is n form the n-th level of the tree T which is also called the n-th generation. For every vertex v at the level n in the tree T there exist a unique path of length n from the root to v. The labels of the vertices on this path form a unique sequence a0 a1 . an in A that corresponds to the vertex v and this sequence is called the full name of v. The correspondence v the full name of v provides a bijection between the vertices in T and the sequences in A. Under this bijection the vertices
