### Abstract

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.

Original language | English |
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Pages (from-to) | 1-26 |

Number of pages | 26 |

Journal | Advances in Applied Mathematics |

Volume | 97 |

DOIs | |

Publication status | Published - 2018 Jun |

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### Keywords

- Continued fractions
- Derangements
- Euler number
- Signed permutations
- Springer number
- Weighted bicolored Motzkin paths

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Advances in Applied Mathematics*,

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