On the Möbius Function of Permutations with One Descent

Abstract

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the Möbius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the Möbius function as a function of the number and positions of pairs of consecutive letters in $\pi$ that are consecutive in value. As a result of this we show that the Möbius function is unbounded on the poset of all permutations. We show that the Möbius function is zero on any interval $[1,\pi]$ where $\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the Möbius function on some other intervals of permutations with at most one descent.