### 摘要

It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length n is the Euler number E_{n}, alternating in sign, if n is odd (even, respectively). Josuat-Vergès obtained a q-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès and Kim. Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Vergès derived bivariate polynomials Q_{n}(t,q) and R_{n}(t,q) as generalized Euler numbers via successive q-derivatives and multiplications by t on polynomials in t. The other goal in this paper is to give a combinatorial interpretation of Q_{n}(t,q) and R_{n}(t,q) as the enumerators of the snakes with restrictions.

原文 | 英語 |
---|---|

頁（從 - 到） | 1-26 |

頁數 | 26 |

期刊 | Advances in Applied Mathematics |

卷 | 97 |

DOIs | |

出版狀態 | 已發佈 - 2018 六月 |

### ASJC Scopus subject areas

- Applied Mathematics

## 指紋 深入研究「Signed countings of types B and D permutations and t,q-Euler numbers」主題。共同形成了獨特的指紋。

## 引用此

*Advances in Applied Mathematics*,

*97*, 1-26. https://doi.org/10.1016/j.aam.2018.02.004