Quasi-Stirling-ap_lap
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\newcommand{\mq}{\mathcal{Q}}
\newcommand{\ap}{{\rm ap\,}}
\newcommand{\lap}{{\rm lap\,}}
$
Quasi-Stirling Permutation, ap, lap 統計量 (back to Data page)
$\bar{\mq}_n$ is the set of all Quasi-Stirling permutation which is a permutation on $\{1^2, 2^2, 3^2, ..., n^2\}$ avoiding (1,2,1,2) and (2,1,2,1)
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 $\pi(0)=0$.
- $F_{lap}^n=\sum_{\rho\in \bar{\mq}_n} x^{\lap(\rho)}, ~
F_{ap}^n=\sum_{\rho\in \bar{\mq}_n} x^{\ap(\rho)}$
Theorem:
$|\bar{\mq}_n| = n!C_n=\frac{(2n)!}{(n+1)!}$
20221024 google colab
目錄：
$ap(\pi)$, for $\pi \in \bar{\mq}_n$
For $F_{ap}^n=\sum_{\rho\in \bar{\mq}_n} x^{\ap(\rho)}$.
$F_{ap}^2(x)=2 x + 2$
$F_{ap}^3(x)=4 x^{2} + 16 x + 10$
$F_{ap}^4(x)=8 x^{3} + 88 x^{2} + 160 x + 80$
$F_{ap}^5(x)=16 x^{4} + 416 x^{3} + 1656 x^{2} + 2096 x + 856$
$F_{ap}^6(x)=32 x^{5} + 1824 x^{4} + 13984 x^{3} + 34144 x^{2} + 33504 x + 11552$
$F_{ap}^7(x)=64 x^{6} + 7680 x^{5} + 105120 x^{4} + 442240 x^{3} + 787200 x^{2} + 631296 x + 188560$
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$\lap(\pi)$ , for $\pi \in \bar{\mq}_n$
$F_{lap}^n=\sum_{\rho\in \bar{\mq}_n} x^{\lap(\rho)}$
$F_{lap}^2(x)=x^{2} + 2 x + 1$
$F_{lap}^3(x)=x^{3} + 10 x^{2} + 13 x + 6$
$F_{lap}^4(x)=x^{4} + 36 x^{3} + 118 x^{2} + 132 x + 49$
$F_{lap}^5(x)=x^{5} + 116 x^{4} + 846 x^{3} + 1856 x^{2} + 1681 x + 540$
$F_{lap}^6(x)=x^{6} + 358 x^{5} + 5279 x^{4} + 20884 x^{3} + 34799 x^{2} + 26278 x + 7441$
$F_{lap}^7(x)=x^{7} + 1086 x^{6} + 30339 x^{5} + 204310 x^{4} + 562815 x^{3} + 752634 x^{2} + 487789 x + 123186$
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