  Let pi = pi_1 pi_2 ... pi_n be a permutation in the symmetric group S_n
written in one-line notation. The pinnacle set of pi, denoted Pin pi, is the
set of all pi_i such that pi_{i-1} < pi_i > pi_{i+1}. This is an analogue of
the well-studied peak set of pi where one considers values rather than
positions. The pinnacle set was introduced 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 S_n
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 pi in S_n.
