2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64980For permutations x and w, let mu(x,w) be the coefficient of highest possible degree in the Kazhdan-Lusztig polynomial P_{x,w}. It is well-known that the coefficients mu(x,w) arise as the edge labels of certain graphs encoding the representations of S_n. The 0-1 Conjecture states that the mu(x,w) are either 0 or 1. We present two counterexamples to this conjecture, the first in S_16, for which x and w are in the same left cell, and the second in S_10. The proof of the counterexample in S_16 relies on computer calculations.15 pages, 4 figures; code for computer calculations included in source packageCombinatoricsRepresentation Theory05E15; 20F55text