Note sur les permutations minimales

The Robinson-Schensted correspondence maps a permutation onto a pair of standard Young tableaux of the same shape. The shape of the two tableaux is referred to as the shape of the permutation. By using the theory of Kazhdan-Luszitg, Hohlweg has recently characterized the permutations with a fixed shape and a minimal inversion number. The present note provides a combinatorial proof of this result by using Viennot's geometric algorithm of the Robinson-Schensted correspondence.