The existence of near-Skolem and hooked near-Skolem sequences
The author has studied the near-Skolem and hooked near-Skolem sequences and it is proved that the arithmetic necessary conditions for the existence of near-Skolem and hooked near-Skolem sequences are also sufficient. These sequences are defined by the following: Let \(m\), \(n\) be integers, \(m\leq n\). A near-Skolem sequence of order \(n\), defect \(m\) is a sequence \(S= (s_ 1,s_ 2,\dots, s_{2n-2})\) of integers \(s_ i\in \{1,2,\dots, m- 1, m+1,\dots,n\}\) which satisfies the following conditions: (1) For every \(k\in \{1,2,\dots, m- 1, m+1,\dots, n\}\) there are exactly two terms \(s_ i\) and \(s_ j\) of \(S\) such that \(s_ i= s_ j= k\). (2) If \(s_ i= s_ j= k\) and \(i< j\) then \(j- i= k\). A hooked near-Skolem sequence of order \(n\), defect \(m\) is a sequence \(HS= (s_ 1, s_ 2,\dots, s_{2n- 1})\) of integers \(s_ i\in \{1,2,\dots, m-1, m+1,\dots, n\}\) which satisfies conditions (1) and (2) and \(s_{2n- 2}= 0\).
scientific article; zbMATH DE number 716465
scientific article