A *Langford pairing* is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of each number k are k units apart. It is well known that such pairings exist for all n = 0 or 3 (mod 4).

For all such n, is there at least one Langford pairing in which the n - 1 numbers between the two appearances of n - 1 are all different? If so, how many?