M-Fold and Extended M-Fold Skolem Sequences

In [2], Stanton and Goulden gave a cyclic construction for Steiner triple systems using a “pairing concept”. They used P (1, n) to denote a set of n pairs of integers in which each of the integers 1 to 2n appears exactly once as an element of a pair and each of the integers 1 to n occurs exactly once as a difference between elements of the same pair. Such a pairing is precisely a Skolem sequence of order n [7]. They used P (1, n)/j to denote a set of n−1 pairs using the integers 1 to 2(n − 1) once each so that each of the integers from 1 to n, except j, occurs as a difference between elements of the same pair exactly once. These pairings are the m-near Skolem sequences discussed in [5]. In [3], Morgan generalized these definitions and used her new pairings to construct balanced ternary designs with block size three. She defined P (1, n) − {j} to be a set of n pairs of integers in which each of the integers from