Alternating permutations and symmetric functions

Details Alternating permutations and symmetric functions Alternating permutations and symmetric functions

2006-03-21

arxiv.org/abs/math/0603520v3

Mathematics

Combinatorics

37 pages, one figure. Correction of gap in the proof of Corollary 6.4, and some further minor corrections

Scientific paper

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.

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