Abstract
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 Ramanujan's “Lost” Notebook.
