On a combinatorial problem Original Research Article

Given a set [n]={1,…,n}, n⩾2, a 2-partition {X,Y} of the set [n] is called a T-partition of [n] when, for every k∈{1,…,n−1}, there exist i∈X and j∈Y such that |i−j|=k, i.e. {X,Y} is a T-partition of [n] iff {|i−j|; i∈X, j∈Y}={1,…,n−1}. In this paper, we are interested in finding the number of T-partitions of a set [n]={1,…,n}, and to describe the asymptotic behavior of these numbers. This problem can be related to the well known Ringel–Kotzig–Rosa conjecture about graceful labelings of a tree (see S.W. Golomb, How to number a graph? in: C.R. Read (Ed.), Graph theory and Computing, Academic Press, New York, 1972, pp. 23–37. A. Rosa, On certain valuations of the vertices of a graph, in: P. Rosenstiehl (Ed.), Théorie de Graphes, Journées Internationales dʹEtude, Rome, 1966, Dunod, Paris, 1967, pp. 349–355), since every graceful labeled tree having n vertices, n⩾2, can be associated with a T-partition of [n]={1,…,n}.