Abstrakt
We show that if a permutation pi contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Mobius function mu[1, pi] of the interval [1, pi] is zero. As a consequence, we prove that the proportion of permutations of length n with principal Mobius function equal to zero is asymptotically bounded below by (1 - 1/e^2) >= 0.3995.
This is the first result determining the value of mu[1, pi] for an asymptotically positive proportion of permutations pi. We further establish other general conditions on a permutation pi that ensure mu[1, pi] = 0.