Multi-ap_lap
定義：
- When $\pi \in \mathcal{S}_M$, where $M$ is $[n]$ repeated $k$ times. The ascent-plateau $ap(\pi)$ is the number of $i$ such that $\pi(i-1)<\pi(i)=\pi(i+1)=...=\pi(i+k-1)$, where $i \in \{1, 2, ..., len(\pi)-1\}$.
- The left ascent-plateau $lap(\pi)$ is the number of $i$ such that $\pi(i-1)<\pi(i)=\pi(i+1)=...=\pi(i+k-1)$, where $i \in \{0, 1, 2, ..., len(\pi)-1\}$ and $\pi(0)=0$.
The statistic of [ap, lap] for $\mathcal{Q}_n$ in link
google colab
目錄：
- $ap(\pi)$, for $\pi \in \mathcal{S}_{ \{1, 1, 2, 2, ...., n, n\} }$
- ( $ap(\pi),lap(\pi)$ ), for $\pi \in \mathcal{S}_{ \{1, 1, 2, 2, ...., n, n\} }$
- $ap(\pi)$, for $\pi \in \mathcal{S}_M$, where $M$ is $[n]$ repeated 3 times.
$ap(\pi)$, for $\pi \in \mathcal{S}_{ \{1, 1, 2, 2, ...., n, n\} }$
For $Ap_n(x)=\sum^n_{i=0}a_{n,i}x^i$, where $a_{n,i}$ is the number of $\pi \in \mathcal{S}_{ \{1, 1, 2, 2, ...., n, n\} }$ such that $ap(\pi)=i$.
$Ap_1(x)=1$,
$Ap_2(x)=4+2 x$,
$Ap_3(x)=58+28x+4x^2$,
$Ap_4(x)=1600+768x+144x^2+8x^3$,
$Ap_5(x)=71296+ 34496 x+ 6936 x^2+ 656 x^3+ 16 x^4$,
$Ap_6(x)= 4675744 + 2274256 x+ 478336 x^2+ 53216 x^3+2816 x^4 + 32 x^5$,
| n | total | numbers | $ap(\pi)$ |
| 1 |
1 |
1 | 0 |
| 2 |
6 |
4 | 0 |
| 2 | 1 |
| 3 |
90 |
58 | 0 |
| 28 | 1 |
| 4 | 2 |
| 4 |
2520 |
1600 | 0 |
| 768 | 1 |
| 144 | 2 |
| 8 | 3 |
| 5 |
113400 |
71296 | 0 |
| 34496 | 1 |
| 6936 | 2 |
| 656 | 3 |
| 16 | 4 |
| 6 |
7484400 |
4675744 | 0 |
| 2274256 | 1 |
| 478336 | 2 |
| 53216 | 3 |
| 2816 | 4 |
| 32 | 5 |
( $ap(\pi),lap(\pi)$ ), for $\pi \in \mathcal{S}_{ \{1, 1, 2, 2, ...., n, n\} }$
| n | total | numbers | $ap(\pi)$ | $lap(\pi)$ |
| 1 |
1 |
1 | 0 | 1 |
| 2 |
6 |
1 | 1 | 2 |
| 1 | 1 | 1 |
| 1 | 0 | 1 |
| 3 | 0 | 0 |
| 3 |
90 |
1 | 2 | 3 |
| 3 | 2 | 2 |
| 7 | 1 | 2 |
| 21 | 1 | 1 |
| 10 | 0 | 1 |
| 48 | 0 | 0 |
| 4 |
2520 |
1 | 3 | 4 |
| 7 | 3 | 3 |
| 29 | 2 | 3 |
| 115 | 2 | 2 |
| 119 | 1 | 2 |
| 649 | 1 | 1 |
| 211 | 0 | 1 |
| 1389 | 0 | 0 |
| 5 |
113400 |
1 | 4 | 5 |
| 15 | 4 | 4 |
| 101 | 3 | 4 |
| 555 | 3 | 3 |
| 951 | 2 | 3 |
| 5985 | 2 | 2 |
| 4091 | 1 | 2 |
| 30405 | 1 | 1 |
| 7456 | 0 | 1 |
| 63840 | 0 | 0 |
| 6 |
7484400 |
1 | 5 | 6 |
| 31 | 5 | 5 |
| 327 | 4 | 5 |
| 2489 | 4 | 4 |
| 6332 | 3 | 4 |
| 46884 | 3 | 3 |
| 51032 | 2 | 3 |
| 427304 | 2 | 2 |
| 217827 | 1 | 2 |
| 2056429 | 1 | 1 |
| 404881 | 0 | 1 |
| 4270863 | 0 | 0 |
$ap(\pi)$, for $\pi \in \mathcal{S}_M$, where $M$ is $[n]$ repeated 3 times.
For $Ap_n(x)=\sum^n_{i=0}a_{n,i}x^i$, where $a_{n,i}$ is the number of $\pi \in \mathcal{S}_M$ such that $ap(\pi)=i$.
$Ap_1(x)= 1 $,
$Ap_2(x)= 17 + 3 x$,
$Ap_3(x)= 1509 + 162 x+ 9 x^2$,
$Ap_4(x)= 340593 + 27801 x+ 1179 x^2+ 27 x^3$,
| n | total | numbers | $ap(\pi)$ |
| 1 |
1 |
1 | 0 |
| 2 |
20 |
17 | 0 |
| 3 | 1 |
| 3 |
1680 |
1509 | 0 |
| 162 | 1 |
| 9 | 2 |
| 4 |
369600 |
340593 | 0 |
| 27801 | 1 |
| 1179 | 2 |
| 27 | 3 |