Alternating permutations and symmetric functions

Authors:
Richard P. Stanley
Affiliations:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2007

Citing 5
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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.