Stirling-ap_lap
定義：
- The ascent-plateau $ap(\pi)$ is the number of $i$ such that $\pi(i-1)<\pi(i)=\pi(i+1)$, where $i \in \{1, 2, ..., len(\pi)-1\}$.
- The left ascent-plateau $lap(\pi)$ is the number of $i$ such that $\pi(i-1)<\pi(i)=\pi(i+1)$, where $i \in \{0, 1, 2, ..., len(\pi)-1\}$ and $\pi(0)=0$.
The statistic of [ap, lap] for $\mathcal{S}_{ \{1, 1, 2, 2, ...., n, n\} }$ in link
google colab
目錄：
$ap(\pi)$, for $\pi \in \mathcal{Q}_n$
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 $,
$Ap_2(x)= 1 + 2 x$,
$Ap_3(x)= 1 + 10 x+ 4 x^2$,
$Ap_4(x)= 1 + 36 x+ 60 x^2+ 8 x^3$,
$Ap_5(x)= 1 + 116 x+ 516 x^2+ 296 x^3+ 16 x^4$,
$Ap_6(x)= 1 + 358 x+ 3508 x^2+ 5168 x^3+ 1328 x^4 + 32 x^5$,
$Ap_7(x)= 1 + 1086 x+ 21120 x^2+ 64240 x^3+ 42960 x^4 + 5664 x^5+ 64 x^6$,
$Ap_8(x)= 1 + 3272 x+ 118632 x^2+ 660880 x^3+ 900560 x^4 + 320064 x^5+ 23488 x^6+ 128 x^7$,
$Ap_8(x)= 1 + 9832 x+ 638968 x^2+ 6049744 x^3+ 14713840 x^4 + 10725184 x^5+ 2225728 x^6+ 95872 x^7+ 256 x^8$,
| n | total | $ap(\pi)$ | numbers |
| 1 |
1 |
1 | 0 |
| 2 |
3 |
1 | 0 |
| 2 | 1 |
| 3 |
15 |
1 | 0 |
| 10 | 1 |
| 4 | 2 |
| 4 |
105 |
1 | 0 |
| 36 | 1 |
| 60 | 2 |
| 8 | 3 |
| 5 |
945 |
1 | 0 |
| 116 | 1 |
| 516 | 2 |
| 296 | 3 |
| 16 | 4 |
| 6 |
10395 |
1 | 0 |
| 358 | 1 |
| 3508 | 2 |
| 5168 | 3 |
| 1328 | 4 |
| 32 | 5 |
| 7 |
135135 |
1 | 0 |
| 1086 | 1 |
| 21120 | 2 |
| 64240 | 3 |
| 42960 | 4 |
| 5664 | 5 |
| 64 | 6 |
| 8 |
2027025 |
1 | 0 |
| 3272 | 1 |
| 118632 | 2 |
| 660880 | 3 |
| 900560 | 4 |
| 320064 | 5 |
| 23488 | 6 |
| 128 | 7 |
| 9 |
34459425 |
1 | 0 |
| 9832 | 1 |
| 638968 | 2 |
| 6049744 | 3 |
| 14713840 | 4 |
| 10725184 | 5 |
| 2225728 | 6 |
| 95872 | 7 |
| 256 | 8 |
( $ap(\pi),lap(\pi)$ ), for $\pi \in \mathcal{Q}_n$
| n | total | numbers | $ap(\pi)$ | $lap(\pi)$ |
| 1 |
1 |
1 | 0 | 1 |
| 2 |
3 |
1 | 1 | 2 |
| 1 | 1 | 1 |
| 1 | 0 | 1 |
| 3 |
15 |
1 | 2 | 3 |
| 3 | 2 | 2 |
| 7 | 1 | 2 |
| 3 | 1 | 1 |
| 1 | 0 | 1 |
| 4 |
105 |
1 | 3 | 4 |
| 7 | 3 | 3 |
| 29 | 2 | 3 |
| 31 | 2 | 2 |
| 29 | 1 | 2 |
| 7 | 1 | 1 |
| 1 | 0 | 1 |
| 5 |
945 |
1 | 4 | 5 |
| 15 | 4 | 4 |
| 101 | 3 | 4 |
| 195 | 3 | 3 |
| 321 | 2 | 3 |
| 195 | 2 | 2 |
| 101 | 1 | 2 |
| 15 | 1 | 1 |
| 1 | 0 | 1 |
| 6 |
10395 |
1 | 5 | 6 |
| 31 | 5 | 5 |
| 327 | 4 | 5 |
| 1001 | 4 | 4 |
| 2507 | 3 | 4 |
| 2661 | 3 | 3 |
| 2507 | 2 | 3 |
| 1001 | 2 | 2 |
| 327 | 1 | 2 |
| 31 | 1 | 1 |
| 1 | 0 | 1 |
| 7 |
135135 |
1 | 6 | 7 |
| 63 | 6 | 6 |
| 1023 | 5 | 6 |
| 4641 | 5 | 5 |
| 16479 | 4 | 5 |
| 26481 | 4 | 4 |
| 37759 | 3 | 4 |
| 26481 | 3 | 3 |
| 16479 | 2 | 3 |
| 4641 | 2 | 2 |
| 1023 | 1 | 2 |
| 63 | 1 | 1 |
| 1 | 0 | 1 |
| 8 |
2027025 |
1 | 7 | 8 |
| 127 | 7 | 7 |
| 3145 | 6 | 7 |
| 20343 | 6 | 6 |
| 98289 | 5 | 6 |
| 221775 | 5 | 5 |
| 439105 | 4 | 5 |
| 461455 | 4 | 4 |
| 439105 | 3 | 4 |
| 221775 | 3 | 3 |
| 98289 | 2 | 3 |
| 20343 | 2 | 2 |
| 3145 | 1 | 2 |
| 127 | 1 | 1 |
| 1 | 0 | 1 |