Berge Sorting Dedicated to Professor Masakazu Kojima on the occasion of his 60th birthday

In 1966, Claude Berge proposed the following sorting problem. Given a string of n alternating white and black pegs on a one-dimensional board consisting of an unlimited number of empty holes, 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. Avis and Deza proved that the alternating string can be sorted in ⌈n 2 ⌉ such Berge 2-moves for n ≥ 5. Extending Berge’s original problem, we consider the same sorting problem using Berge k-moves, i.e., moves which take k adjacent pegs to k vacant adjacent holes. We prove that the alternating string can be sorted in ⌈n 2 ⌉ Berge 3-moves for n 6≡ 0 (mod 4) and in ⌈n 2 ⌉+1 Berge 3-moves for n ≡ 0 (mod 4), for n ≥ 5. In general, we conjecture that, for any k and large enough n, the alternating string can be sorted in ⌈n 2 ⌉ Berge k-moves. This estimate is tight as ⌈n 2 ⌉ is a lower bound for the minimum number of required Berge k-moves for k ≥ 2 and n ≥ 5.