the type B Eulerian numbers with k des (back to Data page)

定義：

• for $\omega \in \mathbb{B}_n$, $des_B(\omega)$ = # $\{1 \leq i \leq n-1 \mid \omega(i) > \omega(i+1) \}$


• $\genfrac{<}{>}{0pt}{}{\mathbb{B}_n}{k} =$ # $\{ \omega \in \mathbb{B}_n \mid des_B(\omega) =k\}$


• sign: for $u \in \mathbb{S}_n, J \in [n]$, let $u^J=\omega \in \mathbb{B}_n$, denote the element given by $\omega(j)=\left\{ \begin{array} {l} u(j), j \notin J\\ -u(j), j\in J \end{array}\right.$.
  $sgn_B(\omega) = (−1)^{|J|} sgn(\omega)$


• $\mathbb{B}_n^+$ denote the set of permutations of positive sign (also known as “even” permutations), and let $\mathbb{B}_n^- = \mathbb{B}_n -\mathbb{B}_n^+$denote the set of permutations with negative sign


• $\genfrac{<}{>}{0pt}{}{\mathbb{B}_n}{k}^+ =$ # $\{ \omega \in \mathbb{B}_n^+ \mid des(\omega) =k\}$, $\genfrac{<}{>}{0pt}{}{\mathbb{B}_n}{k}^- =$ # $\{ \omega \in \mathbb{B}_n^- \mid des(\omega) =k\}$

Triangle of the type B Eulerian numbers $\genfrac{<}{>}{0pt}{}{\mathbb{B}_n}{k}$

Triangle of the type B positive Eulerian numbers $\genfrac{<}{>}{0pt}{}{\mathbb{B}_n}{k}^+$

Triangle of the type B negative Eulerian numbers $\genfrac{<}{>}{0pt}{}{\mathbb{B}_n}{k}^-$

1. [0, 1, 2] , sgn= pos, des= 0
2. [0, -1, 2] , sgn= neg, des= 1
3. [0, 1, -2] , sgn= neg, des= 1
4. [0, -1, -2] , sgn= pos, des= 2
5. [0, 2, 1] , sgn= neg, des= 1
6. [0, -2, 1] , sgn= pos, des= 1
7. [0, 2, -1] , sgn= pos, des= 1
8. [0, -2, -1] , sgn= neg, des= 1