## Abstract

The goal of this paper is to study the asymptotic behavior of almost alternating permutations, that is, permutations that are alternating except for a finite number of exceptions. Let β(l_{1},...,l_{k}) denote the number of permutations which consist of l_{1} ascents, l_{2} descents, l_{3} ascents, and so on. By combining the Viennot triangle and the boustrophedon transform, we obtain the exponential generating function for the numbers β(L, 1^{n-m-1}), where L is a descent-ascent list of size m. As a corollary we have β(L, 1^{n-m-1}) ∼ c(L)·E_{n}, where E_{n} = β(1^{n-1}) denotes the nth Euler number and c(L) is a constant depending on the list L. Using these results and inequalities due to Ehrenborg-Mahajan, we obtain β(1^{a},2,1^{b}) ∼ 2/π·E_{n}, when min(a, b) tends to infinity and where n = a+b+3. From this result we obtain that the asymptotic behavior of β(L_{1}, 1^{a}, L_{2}, 1^{b}, L_{3}) is the product of three constants depending respectively on the lists L_{1}, L_{2}, and L_{3}, times the Euler number E_{a+b+m+1}, where m is the sum of the sizes of the L_{i}'s.

Original language | English |
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Pages (from-to) | 421-437 |

Number of pages | 17 |

Journal | Advances in Applied Mathematics |

Volume | 28 |

Issue number | 3-4 |

DOIs | |

State | Published - 2002 |

### Bibliographical note

Funding Information:The author thanks Margaret Readdy and Doron Zeilberger for inspiring conversations. This research was supported by National Science Foundation, DMS 97-29992, and NEC Research Institute, Inc., while the author was a member of the Institute of Advanced Study. The paper was completed under Swedish Natural Science Research Council Grant DNR 702-238/98 and National Science Foundation, Grants DMS 98-00910 and DMS 99-83660, while visiting Cornell University.

## Keywords

- Boustrophedon transform
- Euler number
- Viennot triangle

## ASJC Scopus subject areas

- Applied Mathematics