On the generalized Berge sorting conjecture

In 1966, Claude Berge proposed the following sorting problem. Given a string of n alternating white and black pegs, rearrange the pegs into a string consisting of ⌈n 2 ⌉ white pegs followed immediately by ⌊n 2 ⌋ black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Berge’s original question was generalized by considering the same sorting problem using only Berge k-moves, i.e., moves which take k adjacent pegs to k vacant adjacent holes. The generalized Berge sorting conjecture states that for any k and large enough n, the alternating string can be sorted in ⌈n 2 ⌉ Berge k-moves. The conjecture holds for k = 2 and n ≥ 5, and for k = 3, n ≥ 5, and n 6≡ 0 (mod 4). We further substantiate this conjecture by showing that it holds for k = 3, n ≥ 20, and n ≡ 0(mod4). The introduced inductive solution generalized previous approaches and could provide insights to tackle the generalized Berge sorting conjecture.