Alternating permutations and symmetric functions

We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,…,n}. These classes include the following: (1) both w and w−1 are alternating, (2) w has certain special shapes, such as (m−1,m−2,…,1), under the RSK algorithm, (3) w has a specified cycle type, and (4) w has a specified number of fixed points. We also enumerate alternating permutations of a multiset. Most of our formulas are umbral expressions where after expanding the expression in powers of a variable E, Ek is interpreted as the Euler number Ek. As a small corollary, we obtain a combinatorial interpretation of the coefficients of an asymptotic expansion appearing in Ramanujan's “Lost” Notebook.