Pinnacle set properties

Abstract

Let π=π1π2…πn be a permutation in the symmetric group Sn written in one-line notation. The pinnacle set of π, denoted Pin π, is the set of all πi such that πi−1<πi>πi+1. This is an analogue of the well-studied peak set of π where one considers values rather than positions. The pinnacle set was so named by Davis, Nelson, Petersen, and Tenner who showed that it has many interesting properties. In particular, they proved that the number of subsets of [n]={1,2,…,n} which can be the pinnacle set of some permutation is a binomial coefficient. Their proof involved a bijection with lattice paths and was somewhat involved. We give a simpler demonstration of this result which does not need lattice paths. Moreover, we show that our map and theirs are different descriptions of the same function. Davis et al. also studied the number of pinnacle sets with maximum m and cardinality d which they denoted by p(m,d). We show that these integers are ballot numbers and give two proofs of this fact: one using finite differences and one bijective. Diaz-Lopez, Harris, Huang, Insko, and Nilsen found a summation formula for calculating the number of permutations in Sn having a given pinnacle set. We derive a new expression for this number which is faster to calculate in many cases. We also show how this method can be adapted to find the number of orderings of a pinnacle set which can be realized by some π∈Sn.