On a combinatorial problem (Q5937446)

A \(T\)-partition of \([n]=\{1, \dots, n\}\), \(n\geq 2\), is a 2-partition \(\{X,Y\}\) where, for every \(k\in\{1, \dots, n-1\}\), there exist \(i\in X\), \(j\in Y\), such that \(|i-j|=k\). The author here addresses the problem of counting the number of elements in \(T_n\), the set of \(T\)-partitions of \([n]\), and checking the asymptotic behavior of \(|T_n|\). Every graceful labeled tree with \(n\) vertices, \(n\geq 2\), can be obtained from a \(T\)-partition of \([n]\) so the work is relevant to the Ringel-Kotzig-Rosa conjecture on graceful labelings of a tree [\textit{S. W. Golomb}, Graph Theory Comput., 23-37 (1972; Zbl 0293.05150)]. The author closes with remarks on the open problem of counting the number of \((s,t)\)-partitions of \([n]\); an \((s,t)\)-partition of \([n]\) is a partition where for three positive integers \(n,s,t\) such that \(s+t\leq n\), \(st\geq n-1\), there exist two disjoint subsets \(X,Y\) of \([n]\), \(|X|=s\), \(|Y|=t\) and \(\{|i-j |: i\in X,j\in Y\}= [n-1]\).