Alternating permutations and symmetric functions
摘要
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.
引用
@article{arxiv.math/0603520,
title = {Alternating permutations and symmetric functions},
author = {Richard P. Stanley},
journal= {arXiv preprint arXiv:math/0603520},
year = {2007}
}
备注
37 pages, one figure. Correction of gap in the proof of Corollary 6.4, and some further minor corrections