Stirling Permutation
Stirling Permutation, asc, des, plat 的統計量 (back to Data page)
定義：
- The Grammar is below, where $x$ is asc, $y$ is des and $z$ is plat
$\begin{equation*}
\left\{
\begin{array}{l}
x \rightarrow x*y*z\\
y \rightarrow x*y*z\\
z \rightarrow x*y*z
\end{array}
\right.,
\end{equation*}$
The statistic of [cc, dc, ds] seems equal to the statistic of [n-1-r, n-1-s, n-1-t], where it is the [asc-1, des-1, plat-1] for the Stirling Permutation
link
Google Colab
目錄：
Grammar result
$
D0 = y\\
$
$
D1 = x y z\\
$
$
D2 = x^{2} y^{2} z + x^{2} y z^{2} + x y^{2} z^{2}\\
$
$
D3 = x^{3} y^{3} z + 4 x^{3} y^{2} z^{2} + x^{3} y z^{3} + 4 x^{2} y^{3} z^{2} + 4 x^{2} y^{2} z^{3} + x y^{3} z^{3}\\
$
$
D4 = x^{4} y^{4} z + 11 x^{4} y^{3} z^{2} + 11 x^{4} y^{2} z^{3} + x^{4} y z^{4} + 11 x^{3} y^{4} z^{2} + 36 x^{3} y^{3} z^{3} + 11 x^{3} y^{2} z^{4} + 11 x^{2} y^{4} z^{3} + 11 x^{2} y^{3} z^{4} + x y^{4} z^{4}\\
$
$
D5 = x^{5} y^{5} z + 26 x^{5} y^{4} z^{2} + 66 x^{5} y^{3} z^{3} + 26 x^{5} y^{2} z^{4} + x^{5} y z^{5} + 26 x^{4} y^{5} z^{2} + 196 x^{4} y^{4} z^{3} + 196 x^{4} y^{3} z^{4} + 26 x^{4} y^{2} z^{5} + 66 x^{3} y^{5} z^{3} + 196 x^{3} y^{4} z^{4} + 66 x^{3} y^{3} z^{5} + 26 x^{2} y^{5} z^{4} + 26 x^{2} y^{4} z^{5} + x y^{5} z^{5}\\
$
$
D6 = x^{6} y^{6} z + 57 x^{6} y^{5} z^{2} + 302 x^{6} y^{4} z^{3} + 302 x^{6} y^{3} z^{4} + 57 x^{6} y^{2} z^{5} + x^{6} y z^{6} + 57 x^{5} y^{6} z^{2} + 848 x^{5} y^{5} z^{3} + 1898 x^{5} y^{4} z^{4} + 848 x^{5} y^{3} z^{5} + 57 x^{5} y^{2} z^{6} + 302 x^{4} y^{6} z^{3} + 1898 x^{4} y^{5} z^{4} + 1898 x^{4} y^{4} z^{5} + 302 x^{4} y^{3} z^{6} + 302 x^{3} y^{6} z^{4} + 848 x^{3} y^{5} z^{5} + 302 x^{3} y^{4} z^{6} + 57 x^{2} y^{6} z^{5} + 57 x^{2} y^{5} z^{6} + x y^{6} z^{6}\\
$
$
D7 = x^{7} y^{7} z + 120 x^{7} y^{6} z^{2} + 1191 x^{7} y^{5} z^{3} + 2416 x^{7} y^{4} z^{4} + 1191 x^{7} y^{3} z^{5} + 120 x^{7} y^{2} z^{6} + x^{7} y z^{7} + 120 x^{6} y^{7} z^{2} + 3228 x^{6} y^{6} z^{3} + 13644 x^{6} y^{5} z^{4} + 13644 x^{6} y^{4} z^{5} + 3228 x^{6} y^{3} z^{6} + 120 x^{6} y^{2} z^{7} + 1191 x^{5} y^{7} z^{3} + 13644 x^{5} y^{6} z^{4} + 28470 x^{5} y^{5} z^{5} + 13644 x^{5} y^{4} z^{6} + 1191 x^{5} y^{3} z^{7} + 2416 x^{4} y^{7} z^{4} + 13644 x^{4} y^{6} z^{5} + 13644 x^{4} y^{5} z^{6} + 2416 x^{4} y^{4} z^{7} + 1191 x^{3} y^{7} z^{5} + 3228 x^{3} y^{6} z^{6} + 1191 x^{3} y^{5} z^{7} + 120 x^{2} y^{7} z^{6} + 120 x^{2} y^{6} z^{7} + x y^{7} z^{7}\\
$
$
D8 = x^{8} y^{8} z + 247 x^{8} y^{7} z^{2} + 4293 x^{8} y^{6} z^{3} + 15619 x^{8} y^{5} z^{4} + 15619 x^{8} y^{4} z^{5} + 4293 x^{8} y^{3} z^{6} + 247 x^{8} y^{2} z^{7} + x^{8} y z^{8} + 247 x^{7} y^{8} z^{2} + 11364 x^{7} y^{7} z^{3} + 82281 x^{7} y^{6} z^{4} + 153352 x^{7} y^{5} z^{5} + 82281 x^{7} y^{4} z^{6} + 11364 x^{7} y^{3} z^{7} + 247 x^{7} y^{2} z^{8} + 4293 x^{6} y^{8} z^{3} + 82281 x^{6} y^{7} z^{4} + 306078 x^{6} y^{6} z^{5} + 306078 x^{6} y^{5} z^{6} + 82281 x^{6} y^{4} z^{7} + 4293 x^{6} y^{3} z^{8} + 15619 x^{5} y^{8} z^{4} + 153352 x^{5} y^{7} z^{5} + 306078 x^{5} y^{6} z^{6} + 153352 x^{5} y^{5} z^{7} + 15619 x^{5} y^{4} z^{8} + 15619 x^{4} y^{8} z^{5} + 82281 x^{4} y^{7} z^{6} + 82281 x^{4} y^{6} z^{7} + 15619 x^{4} y^{5} z^{8} + 4293 x^{3} y^{8} z^{6} + 11364 x^{3} y^{7} z^{7} + 4293 x^{3} y^{6} z^{8} + 247 x^{2} y^{8} z^{7} + 247 x^{2} y^{7} z^{8} + x y^{8} z^{8}\\
$
$
D9 = x^{9} y^{9} z + 502 x^{9} y^{8} z^{2} + 14608 x^{9} y^{7} z^{3} + 88234 x^{9} y^{6} z^{4} + 156190 x^{9} y^{5} z^{5} + 88234 x^{9} y^{4} z^{6} + 14608 x^{9} y^{3} z^{7} + 502 x^{9} y^{2} z^{8} + x^{9} y z^{9} + 502 x^{8} y^{9} z^{2} + 38044 x^{8} y^{8} z^{3} + 443016 x^{8} y^{7} z^{4} + 1385398 x^{8} y^{6} z^{5} + 1385398 x^{8} y^{5} z^{6} + 443016 x^{8} y^{4} z^{7} + 38044 x^{8} y^{3} z^{8} + 502 x^{8} y^{2} z^{9} + 14608 x^{7} y^{9} z^{3} + 443016 x^{7} y^{8} z^{4} + 2682324 x^{7} y^{7} z^{5} + 4746400 x^{7} y^{6} z^{6} + 2682324 x^{7} y^{5} z^{7} + 443016 x^{7} y^{4} z^{8} + 14608 x^{7} y^{3} z^{9} + 88234 x^{6} y^{9} z^{4} + 1385398 x^{6} y^{8} z^{5} + 4746400 x^{6} y^{7} z^{6} + 4746400 x^{6} y^{6} z^{7} + 1385398 x^{6} y^{5} z^{8} + 88234 x^{6} y^{4} z^{9} + 156190 x^{5} y^{9} z^{5} + 1385398 x^{5} y^{8} z^{6} + 2682324 x^{5} y^{7} z^{7} + 1385398 x^{5} y^{6} z^{8} + 156190 x^{5} y^{5} z^{9} + 88234 x^{4} y^{9} z^{6} + 443016 x^{4} y^{8} z^{7} + 443016 x^{4} y^{7} z^{8} + 88234 x^{4} y^{6} z^{9} + 14608 x^{3} y^{9} z^{7} + 38044 x^{3} y^{8} z^{8} + 14608 x^{3} y^{7} z^{9} + 502 x^{2} y^{9} z^{8} + 502 x^{2} y^{8} z^{9} + x y^{9} z^{9}\\
$
$
D10 = x^{10} y^{10} z + 1013 x^{10} y^{9} z^{2} + 47840 x^{10} y^{8} z^{3} + 455192 x^{10} y^{7} z^{4} + 1310354 x^{10} y^{6} z^{5} + 1310354 x^{10} y^{5} z^{6} + 455192 x^{10} y^{4} z^{7} + 47840 x^{10} y^{3} z^{8} + 1013 x^{10} y^{2} z^{9} + x^{10} y z^{10} + 1013 x^{9} y^{10} z^{2} + 123168 x^{9} y^{9} z^{3} + 2207888 x^{9} y^{8} z^{4} + 10822208 x^{9} y^{7} z^{5} + 18030486 x^{9} y^{6} z^{6} + 10822208 x^{9} y^{5} z^{7} + 2207888 x^{9} y^{4} z^{8} + 123168 x^{9} y^{3} z^{9} + 1013 x^{9} y^{2} z^{10} + 47840 x^{8} y^{10} z^{3} + 2207888 x^{8} y^{9} z^{4} + 20499876 x^{8} y^{8} z^{5} + 58337852 x^{8} y^{7} z^{6} + 58337852 x^{8} y^{6} z^{7} + 20499876 x^{8} y^{5} z^{8} + 2207888 x^{8} y^{4} z^{9} + 47840 x^{8} y^{3} z^{10} + 455192 x^{7} y^{10} z^{4} + 10822208 x^{7} y^{9} z^{5} + 58337852 x^{7} y^{8} z^{6} + 99674400 x^{7} y^{7} z^{7} + 58337852 x^{7} y^{6} z^{8} + 10822208 x^{7} y^{5} z^{9} + 455192 x^{7} y^{4} z^{10} + 1310354 x^{6} y^{10} z^{5} + 18030486 x^{6} y^{9} z^{6} + 58337852 x^{6} y^{8} z^{7} + 58337852 x^{6} y^{7} z^{8} + 18030486 x^{6} y^{6} z^{9} + 1310354 x^{6} y^{5} z^{10} + 1310354 x^{5} y^{10} z^{6} + 10822208 x^{5} y^{9} z^{7} + 20499876 x^{5} y^{8} z^{8} + 10822208 x^{5} y^{7} z^{9} + 1310354 x^{5} y^{6} z^{10} + 455192 x^{4} y^{10} z^{7} + 2207888 x^{4} y^{9} z^{8} + 2207888 x^{4} y^{8} z^{9} + 455192 x^{4} y^{7} z^{10} + 47840 x^{3} y^{10} z^{8} + 123168 x^{3} y^{9} z^{9} + 47840 x^{3} y^{8} z^{10} + 1013 x^{2} y^{10} z^{9} + 1013 x^{2} y^{9} z^{10} + x y^{10} z^{10}\\
$
Data
| n | numbers | permutation | asc | des | plat |
| 2 |
1 | [1, 2, 2, 1] | 1 | 1 | 1 |
| 1 | [1, 1, 2, 2] | 1 | 0 | 2 |
| 1 | [2, 2, 1, 1] | 0 | 1 | 2 |
| 3 |
1 | [1, 2, 3, 3, 2, 1] | 2 | 2 | 1 |
| 4 | [1, 2, 2, 3, 3, 1] | 2 | 1 | 2 |
| [1, 2, 2, 1, 3, 3] |
| [1, 3, 3, 1, 2, 2] |
| [1, 1, 2, 3, 3, 2] |
| 1 | [1, 1, 2, 2, 3, 3] | 2 | 0 | 3 |
| 4 | [2, 3, 3, 2, 1, 1] | 1 | 2 | 2 |
| [2, 2, 1, 3, 3, 1] |
| [3, 3, 1, 2, 2, 1] |
| [1, 3, 3, 2, 2, 1] |
| 4 | [2, 2, 3, 3, 1, 1] | 1 | 1 | 3 |
| [2, 2, 1, 1, 3, 3] |
| [3, 3, 1, 1, 2, 2] |
| [1, 1, 3, 3, 2, 2] |
| 1 | [3, 3, 2, 2, 1, 1] | 0 | 2 | 3 |
| 4 |
1 | [1, 2, 3, 4, 4, 3, 2, 1] | 3 | 3 | 1 |
| 11 | [1, 2, 3, 3, 4, 4, 2, 1] | 3 | 2 | 2 |
| [1, 2, 3, 3, 2, 4, 4, 1] |
| [1, 2, 3, 3, 2, 1, 4, 4] |
| [1, 2, 4, 4, 2, 3, 3, 1] |
| [1, 2, 2, 3, 4, 4, 3, 1] |
| [1, 2, 4, 4, 2, 1, 3, 3] |
| [1, 2, 2, 1, 3, 4, 4, 3] |
| [1, 3, 4, 4, 3, 1, 2, 2] |
| [1, 3, 3, 1, 2, 4, 4, 2] |
| [1, 4, 4, 1, 2, 3, 3, 2] |
| [1, 1, 2, 3, 4, 4, 3, 2] |
| 11 | [1, 2, 2, 3, 3, 4, 4, 1] | 3 | 1 | 3 |
| [1, 2, 2, 3, 3, 1, 4, 4] |
| [1, 2, 2, 4, 4, 1, 3, 3] |
| [1, 2, 2, 1, 3, 3, 4, 4] |
| [1, 3, 3, 4, 4, 1, 2, 2] |
| [1, 3, 3, 1, 2, 2, 4, 4] |
| [1, 1, 2, 3, 3, 4, 4, 2] |
| [1, 1, 2, 3, 3, 2, 4, 4] |
| [1, 4, 4, 1, 2, 2, 3, 3] |
| [1, 1, 2, 4, 4, 2, 3, 3] |
| [1, 1, 2, 2, 3, 4, 4, 3] |
| 1 | [1, 1, 2, 2, 3, 3, 4, 4] | 3 | 0 | 4 |
| 11 | [2, 3, 4, 4, 3, 2, 1, 1] | 2 | 3 | 2 |
| [2, 3, 3, 2, 1, 4, 4, 1] |
| [2, 4, 4, 2, 1, 3, 3, 1] |
| [2, 2, 1, 3, 4, 4, 3, 1] |
| [3, 4, 4, 3, 1, 2, 2, 1] |
| [3, 3, 1, 2, 4, 4, 2, 1] |
| [1, 3, 4, 4, 3, 2, 2, 1] |
| [1, 3, 3, 2, 4, 4, 2, 1] |
| [4, 4, 1, 2, 3, 3, 2, 1] |
| [1, 4, 4, 2, 3, 3, 2, 1] |
| [1, 2, 4, 4, 3, 3, 2, 1] |
| 36 | [2, 3, 3, 4, 4, 2, 1, 1] | 2 | 2 | 3 |
| [2, 3, 3, 2, 4, 4, 1, 1] |
| [2, 3, 3, 2, 1, 1, 4, 4] |
| [2, 4, 4, 2, 3, 3, 1, 1] |
| [2, 2, 3, 4, 4, 3, 1, 1] |
| [2, 2, 3, 3, 1, 4, 4, 1] |
| [2, 2, 4, 4, 1, 3, 3, 1] |
| [2, 2, 1, 3, 3, 4, 4, 1] |
| [2, 2, 1, 3, 3, 1, 4, 4] |
| [2, 4, 4, 2, 1, 1, 3, 3] |
| [2, 2, 1, 4, 4, 1, 3, 3] |
| [2, 2, 1, 1, 3, 4, 4, 3] |
| [3, 3, 4, 4, 1, 2, 2, 1] |
| [3, 3, 1, 2, 2, 4, 4, 1] |
| [3, 3, 1, 2, 2, 1, 4, 4] |
| [1, 3, 3, 4, 4, 2, 2, 1] |
| [1, 3, 3, 2, 2, 4, 4, 1] |
| [1, 3, 3, 2, 2, 1, 4, 4] |
| [4, 4, 1, 2, 2, 3, 3, 1] |
| [1, 4, 4, 2, 2, 3, 3, 1] |
| [1, 2, 2, 4, 4, 3, 3, 1] |
| [4, 4, 1, 2, 2, 1, 3, 3] |
| [1, 4, 4, 2, 2, 1, 3, 3] |
| [1, 2, 2, 1, 4, 4, 3, 3] |
| [3, 4, 4, 3, 1, 1, 2, 2] |
| [3, 3, 1, 4, 4, 1, 2, 2] |
| [3, 3, 1, 1, 2, 4, 4, 2] |
| [4, 4, 1, 3, 3, 1, 2, 2] |
| [1, 4, 4, 3, 3, 1, 2, 2] |
| [1, 3, 3, 1, 4, 4, 2, 2] |
| [1, 4, 4, 1, 3, 3, 2, 2] |
| [1, 1, 3, 4, 4, 3, 2, 2] |
| [1, 1, 3, 3, 2, 4, 4, 2] |
| [4, 4, 1, 1, 2, 3, 3, 2] |
| [1, 1, 4, 4, 2, 3, 3, 2] |
| [1, 1, 2, 4, 4, 3, 3, 2] |
| 11 | [2, 2, 3, 3, 4, 4, 1, 1] | 2 | 1 | 4 |
| [2, 2, 3, 3, 1, 1, 4, 4] |
| [2, 2, 4, 4, 1, 1, 3, 3] |
| [2, 2, 1, 1, 3, 3, 4, 4] |
| [3, 3, 4, 4, 1, 1, 2, 2] |
| [3, 3, 1, 1, 2, 2, 4, 4] |
| [1, 1, 3, 3, 4, 4, 2, 2] |
| [1, 1, 3, 3, 2, 2, 4, 4] |
| [4, 4, 1, 1, 2, 2, 3, 3] |
| [1, 1, 4, 4, 2, 2, 3, 3] |
| [1, 1, 2, 2, 4, 4, 3, 3] |
| 11 | [3, 4, 4, 3, 2, 2, 1, 1] | 1 | 3 | 3 |
| [3, 3, 2, 4, 4, 2, 1, 1] |
| [3, 3, 2, 2, 1, 4, 4, 1] |
| [4, 4, 2, 3, 3, 2, 1, 1] |
| [2, 4, 4, 3, 3, 2, 1, 1] |
| [4, 4, 2, 2, 1, 3, 3, 1] |
| [2, 2, 1, 4, 4, 3, 3, 1] |
| [4, 4, 3, 3, 1, 2, 2, 1] |
| [3, 3, 1, 4, 4, 2, 2, 1] |
| [4, 4, 1, 3, 3, 2, 2, 1] |
| [1, 4, 4, 3, 3, 2, 2, 1] |
| 11 | [3, 3, 4, 4, 2, 2, 1, 1] | 1 | 2 | 4 |
| [3, 3, 2, 2, 4, 4, 1, 1] |
| [3, 3, 2, 2, 1, 1, 4, 4] |
| [4, 4, 2, 2, 3, 3, 1, 1] |
| [2, 2, 4, 4, 3, 3, 1, 1] |
| [4, 4, 2, 2, 1, 1, 3, 3] |
| [2, 2, 1, 1, 4, 4, 3, 3] |
| [4, 4, 3, 3, 1, 1, 2, 2] |
| [3, 3, 1, 1, 4, 4, 2, 2] |
| [4, 4, 1, 1, 3, 3, 2, 2] |
| [1, 1, 4, 4, 3, 3, 2, 2] |
| 1 | [4, 4, 3, 3, 2, 2, 1, 1] | 0 | 3 | 4 |