Excedance Number of Derangement in Multi-Set (back to Data page)

定義：

• $M=\{ 1^{a_1}, 2^{a_2}, ..., n^{a_n}\}$ is a multi-set, where $a_i \in \mathbb{N}$ and 1~n repeated a times n $M$.


• $N=\sum^n_{i=1}i\cdot a_i$


• $\pi = \pi_1\pi_2 \cdots \pi_{N}$ is one line permutation in $\mathcal{S}_M$


• The excedance set $EXC(\pi)= \Big| \{ i \in [N-1] \mid \pi_i > i \} \Big|$


• $exc(\pi)$= $ \Big| EXC(\pi) \Big|$




• $\pi = \pi_1\pi_2 \cdots \pi_{N}$ is called a derangement in $\mathcal{S}_M$ if $\pi_i \neq i$ for all $i$

目錄：

• $a_1=a_2=...=a_n=2$
• $a_1=3, a_2=...=a_n=2$
• $a_1=2, a_2=3, a_3=...=a_n=2$
• $a_{odd}=2, a_{even}=3$
• $a_1=a_2=...=a_n=3$
• $a_1=2, a_2=...=a_n=3$
• $a_1=3, a_2=2, a_3=...=a_n=3$
• $a_{odd}=3, a_{even}=2$

$a_1=a_2=...=a_n=2$

$a_1=3, a_2=...=a_n=2$

$a_1=2, a_2=3, a_3=...=a_n=2$

$a_{odd}=2, a_{even}=3$

$a_1=a_2=...=a_n=3$

$a_1=2, a_2=...=a_n=3$

$a_1=3, a_2=2, a_3=...=a_n=3$

$a_{odd}=3, a_{even}=2$