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, E k is interpreted as the Euler number E k . As a small corollary, we obtain a combinatorial interpretation of the coefficients of an asymptotic expansion appearing in Ramanujanʹs “Lost” Notebook.