Eulerian numbers with k des (back to Data page)

定義：

• des($\omega$) = # $\{1 \leq i \leq n-1 \mid \omega(i) > \omega(i+1) \}$


• $\genfrac{<}{>}{0pt}{}{n}{k} =$ # $\{ \omega \in \mathbb{S}_n \mid des(\omega) =k-1\}$


• sign: The sign of a permutation is 1 if it can be written as a product of an even number of transpositions; otherwise the sign is −1. The sign of a permutation is well-defined, and denoted by sgn($\omega$). It is also related to other statistics for permutations, e.g., $sgn(\omega) = (−1)^{n−c(\omega)} = (−1)^{inv(\omega)}$ , where $c(\omega)$ is the number of cycles of $\omega$, and $inv(\omega)$ is the number of inversions of $\omega$.


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


• $\genfrac{<}{>}{0pt}{}{n}{k}^+ =$ # $\{ \omega \in \mathbb{S}_n^+ \mid des(\omega) =k-1\}$, $\genfrac{<}{>}{0pt}{}{n}{k}^- =$ # $\{ \omega \in \mathbb{S}_n^- \mid des(\omega) =k-1\}$

Triangle of the Eulerian numbers $\genfrac{<}{>}{0pt}{}{n}{k}$

Triangle of the positive Eulerian numbers $\genfrac{<}{>}{0pt}{}{n}{k}^+$

Triangle of the negative Eulerian numbers $\genfrac{<}{>}{0pt}{}{n}{k}^-$