Stirling Permutation
(1)-Stirling Permutation, asc, des, plat 的統計量 (back to Data page)
定義：
- (1)-Stirling Permutation $\mathcal{Q}^{(1)}_n$ is Stirling Permutation on $\{1, 2^2, 3^3, ..., n^2\}$.
Google Colab
$$
C_2(x,y,z)=x^{2} y z + x y^{2} z
$$
\begin{align*}
z &: x^{2} y + x y^{2}
\end{align*}
$$
C_3(x,y,z)=x^{3} y^{2} z + x^{3} y z^{2} + x^{2} y^{3} z + 4 x^{2} y^{2} z^{2} + x y^{3} z^{2}
$$
\begin{align*}
z^2 &: x^{3} y + 4 x^{2} y^{2} + x y^{3}\\
z &: x^{3} y^{2} + x^{2} y^{3}
\end{align*}
$$
C_4(x,y,z)=x^{4} y^{3} z + 4 x^{4} y^{2} z^{2} + x^{4} y z^{3} + x^{3} y^{4} z + 14 x^{3} y^{3} z^{2} + 11 x^{3} y^{2} z^{3} + 4 x^{2} y^{4} z^{2} + 11 x^{2} y^{3} z^{3} + x y^{4} z^{3}
$$
\begin{align*}
z^3 &: x^{4} y + 11 x^{3} y^{2} + 11 x^{2} y^{3} + x y^{4}\\
z^2 &: 4 x^{4} y^{2} + 14 x^{3} y^{3} + 4 x^{2} y^{4}\\
z &: x^{4} y^{3} + x^{3} y^{4}
\end{align*}
$$
C_5(x,y,z)=x^{5} y^{4} z + 11 x^{5} y^{3} z^{2} + 11 x^{5} y^{2} z^{3} + x^{5} y z^{4} + x^{4} y^{5} z + 36 x^{4} y^{4} z^{2} + 91 x^{4} y^{3} z^{3} + 26 x^{4} y^{2} z^{4} + 11 x^{3} y^{5} z^{2} + 91 x^{3} y^{4} z^{3} + 66 x^{3} y^{3} z^{4} + 11 x^{2} y^{5} z^{3} + 26 x^{2} y^{4} z^{4} + x y^{5} z^{4}
$$
\begin{align*}
z^4 &: x^{5} y + 26 x^{4} y^{2} + 66 x^{3} y^{3} + 26 x^{2} y^{4} + x y^{5}\\
z^3 &: 11 x^{5} y^{2} + 91 x^{4} y^{3} + 91 x^{3} y^{4} + 11 x^{2} y^{5}\\
z^2 &: 11 x^{5} y^{3} + 36 x^{4} y^{4} + 11 x^{3} y^{5}\\
z &: x^{5} y^{4} + x^{4} y^{5}
\end{align*}
$$
C_6(x,y,z)=x^{6} y^{5} z + 26 x^{6} y^{4} z^{2} + 66 x^{6} y^{3} z^{3} + 26 x^{6} y^{2} z^{4} + x^{6} y z^{5} + x^{5} y^{6} z + 82 x^{5} y^{5} z^{2} + 472 x^{5} y^{4} z^{3} + 432 x^{5} y^{3} z^{4} + 57 x^{5} y^{2} z^{5} + 26 x^{4} y^{6} z^{2} + 472 x^{4} y^{5} z^{3} + 992 x^{4} y^{4} z^{4} + 302 x^{4} y^{3} z^{5} + 66 x^{3} y^{6} z^{3} + 432 x^{3} y^{5} z^{4} + 302 x^{3} y^{4} z^{5} + 26 x^{2} y^{6} z^{4} + 57 x^{2} y^{5} z^{5} + x y^{6} z^{5}
$$
\begin{align*}
z^5 &: x^{6} y + 57 x^{5} y^{2} + 302 x^{4} y^{3} + 302 x^{3} y^{4} + 57 x^{2} y^{5} + x y^{6}\\
z^4 &: 26 x^{6} y^{2} + 432 x^{5} y^{3} + 992 x^{4} y^{4} + 432 x^{3} y^{5} + 26 x^{2} y^{6}\\
z^3 &: 66 x^{6} y^{3} + 472 x^{5} y^{4} + 472 x^{4} y^{5} + 66 x^{3} y^{6}\\
z^2 &: 26 x^{6} y^{4} + 82 x^{5} y^{5} + 26 x^{4} y^{6}\\
z &: x^{6} y^{5} + x^{5} y^{6}
\end{align*}
$$
C_7(x,y,z)=x^{7} y^{6} z + 57 x^{7} y^{5} z^{2} + 302 x^{7} y^{4} z^{3} + 302 x^{7} y^{3} z^{4} + 57 x^{7} y^{2} z^{5} + x^{7} y z^{6} + x^{6} y^{7} z + 176 x^{6} y^{6} z^{2} + 1982 x^{6} y^{5} z^{3} + 4012 x^{6} y^{4} z^{4} + 1737 x^{6} y^{3} z^{5} + 120 x^{6} y^{2} z^{6} + 57 x^{5} y^{7} z^{2} + 1982 x^{5} y^{6} z^{3} + 8688 x^{5} y^{5} z^{4} + 7638 x^{5} y^{4} z^{5} + 1191 x^{5} y^{3} z^{6} + 302 x^{4} y^{7} z^{3} + 4012 x^{4} y^{6} z^{4} + 7638 x^{4} y^{5} z^{5} + 2416 x^{4} y^{4} z^{6} + 302 x^{3} y^{7} z^{4} + 1737 x^{3} y^{6} z^{5} + 1191 x^{3} y^{5} z^{6} + 57 x^{2} y^{7} z^{5} + 120 x^{2} y^{6} z^{6} + x y^{7} z^{6}
$$
\begin{align*}
z^6 &: x^{7} y + 120 x^{6} y^{2} + 1191 x^{5} y^{3} + 2416 x^{4} y^{4} + 1191 x^{3} y^{5} + 120 x^{2} y^{6} + x y^{7}\\
z^5 &: 57 x^{7} y^{2} + 1737 x^{6} y^{3} + 7638 x^{5} y^{4} + 7638 x^{4} y^{5} + 1737 x^{3} y^{6} + 57 x^{2} y^{7}\\
z^4 &: 302 x^{7} y^{3} + 4012 x^{6} y^{4} + 8688 x^{5} y^{5} + 4012 x^{4} y^{6} + 302 x^{3} y^{7}\\
z^3 &: 302 x^{7} y^{4} + 1982 x^{6} y^{5} + 1982 x^{5} y^{6} + 302 x^{4} y^{7}\\
z^2 &: 57 x^{7} y^{5} + 176 x^{6} y^{6} + 57 x^{5} y^{7}\\
z &: x^{7} y^{6} + x^{6} y^{7}
\end{align*}