Quasi-Stirling Permutation $ \newcommand{\mq}{\mathcal{Q}} \newcommand{\asc}{{\rm asc\,}} \newcommand{\des}{{\rm des\,}} \newcommand{\plat}{{\rm plat\,}} \newcommand{\ap}{{\rm ap\,}} \newcommand{\lap}{{\rm lap\,}} $

Quasi-Stirling Permutation, asc, des, plat 的統計量 (back to Data page)

定義：

$\bar{\mq}_n$ 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)

Theorem: $|\bar{\mq}_n| = n!C_n=\frac{(2n)!}{(n+1)!}$

20221024 google colab

$$F_{asc}^n=\sum_{\rho\in \bar{\mq}_n} x^{\asc(\rho)}, ~ F_{plt}^n=\sum_{\rho\in \bar{\mq}_n} x^{\plat(\rho)}, ~ F_{des}^n=\sum_{\rho\in \bar{\mq}_n} x^{\des(\rho)}, ~$$ $F_{asc}^2(x)=3 x + 1$
$F_{plt}^2(x)=2 x^{2} + 2 x$
$F_{des}^2(x)=3 x + 1$

$F_{asc}^3(x)=16 x^{2} + 13 x + 1$
$F_{plt}^3(x)=6 x^{3} + 18 x^{2} + 6 x$
$F_{des}^3(x)=16 x^{2} + 13 x + 1$

$F_{asc}^4(x)=125 x^{3} + 171 x^{2} + 39 x + 1$
$F_{plt}^4(x)=24 x^{4} + 144 x^{3} + 144 x^{2} + 24 x$
$F_{des}^4(x)=125 x^{3} + 171 x^{2} + 39 x + 1$

$F_{asc}^5(x)=1296 x^{4} + 2551 x^{3} + 1091 x^{2} + 101 x + 1$
$F_{plt}^5(x)=120 x^{5} + 1200 x^{4} + 2400 x^{3} + 1200 x^{2} + 120 x$
$F_{des}^5(x)=1296 x^{4} + 2551 x^{3} + 1091 x^{2} + 101 x + 1$

$F_{asc}^6(x)=16807 x^{5} + 43653 x^{4} + 28838 x^{3} + 5498 x^{2} + 243 x + 1$
$F_{plt}^6(x)=720 x^{6} + 10800 x^{5} + 36000 x^{4} + 36000 x^{3} + 10800 x^{2} + 720 x$
$F_{des}^6(x)=16807 x^{5} + 43653 x^{4} + 28838 x^{3} + 5498 x^{2} + 243 x + 1$