Alternating permutations and symmetric functions

2006/03/21 by Richard P. Stanley, Stanley, Richard P. · 1 citation
Mathematics · #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.math/0603520

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.

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