On the Möbius Function of Permutations with One Descent

Jason P. Smith

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.

Keywords


Möbius function

Full Text: PDF