A set of necessary conditions for the existence of a partition of \(\{1, \ldots, 2m – 1, L\}\) into differences \(d, d + 1, \ldots, d + m – 1\) is \((m, L) \equiv (0, 0), (1, d + 1), (2, 1), (3, d) \pmod{(4, 2)}\) and \(m \geq 2d – 2\) or \(m = 1\). If \(m = 2d – 2\) then \(L = 5d – 5\), if \(m = 2d – 1\) then \(4d – 2 \leq L \leq 6d – 4\) and if \(m \geq 2d\) then \(2m \leq L \leq 3m + d – 2\). Similar conditions for the partition of \(\{1, \ldots, 2m, L\} \setminus \{2\}\) into differences \(d, d + 1, \ldots, d + m – 1\) are \((m, L) \equiv (0, 0), (1, d + 1), (2, 1), (3, d) \pmod{(4, 2)}\), \((d, m, L) \neq (1, 1, 4), (2, 3, 8)\) and \(m \geq 2d – 2\), \(m = 1\) or \((d, m, L) = (3, 2, 7), (3, 3, 9)\). If \(m = 2d – 2\) then \(L = 5d – 5, 5d – 3\), if \(m = 2d – 1\) then \(4d – 1 \leq L \leq 6d – 2\) and if \(m \geq 2d\) then \(2m + 1 \leq L \leq 3m + d – 1\).
It is shown that for many cases when the necessary conditions hold, the set \(\{1, \ldots, 2m – 1, L\}\) and \(\{1, \ldots, 2m – 1, L\} \setminus \{2\}\) can be so partitioned. These partitions exist for all the minimum and maximum \(L\), when \(d \leq 3\), when \(m = 1\) and when \(m \geq 8d – 3\) (\(m \geq 8d + 4\) in the hooked case). The constructions given fully solve the existence of these partitions if the necessary conditions for the existence of extended and hooked extended Langford sequences are sufficient.