permB_desA_desB_neg
Permutaion of type B, $\pi(0)=0$, in words, with des$_A$, des$_B$, negative (back to Data page)
定義：
- Permutaion of type B：
- $des_A$($\rho$) = # $\{i \in [n-1] \mid \rho(i) > \rho(i+1) \}$
- $des_B$($\rho$) = # $\{i \in [0, n-1] \mid \rho(i) > \rho(i+1) \}$
- $neg$($\rho$) = # $\{i \mid \rho(i) < 0 \}$
$$F_n(x,y,z)=\sum_{\sigma \in S^B_n} x^{des_A(\sigma)+1} y^{des_B(\sigma)} z^{neg(\sigma)}=\sum_{\pi \in Q^{(1)}_n} x^{lap(\pi)} y^{ap(\pi)} z^{even(\pi)}$$
check
(1)-Stirling Permutation, $\mathcal{Q}^{(1)}_n$, lap, ap, even
google colab
$$
F_1=x y z + x
$$
$$
F_2=x^{2} y^{2} z^{2} + 2 x^{2} y z + x^{2} y + x y z^{2} + 2 x y z + x
$$
$$
z^{2}:
\begin{bmatrix}
0 & 0 & 0 & \\
0 & 1 & 0 & \\
0 & 0 & 1 & \\
\end{bmatrix}
,~
z:
\begin{bmatrix}
0 & 0 & \\
0 & 2 & \\
0 & 2 & \\
\end{bmatrix}
,~
1:
\begin{bmatrix}
0 & 0 & \\
1 & 0 & \\
0 & 1 & \\
\end{bmatrix}
$$
$$
F_3=x^{3} y^{3} z^{3} + 3 x^{3} y^{2} z^{2} + 3 x^{3} y^{2} z + x^{3} y^{2} + 4 x^{2} y^{2} z^{3} + 9 x^{2} y^{2} z^{2} + 3 x^{2} y^{2} z + 3 x^{2} y z^{2} + 9 x^{2} y z + 4 x^{2} y + x y z^{3} + 3 x y z^{2} + 3 x y z + x
$$
$$
z^{3}:
\begin{bmatrix}
0 & 0 & 0 & 0 & \\
0 & 1 & 0 & 0 & \\
0 & 0 & 4 & 0 & \\
0 & 0 & 0 & 1 & \\
\end{bmatrix}
,~
z^{2}:
\begin{bmatrix}
0 & 0 & 0 & \\
0 & 3 & 0 & \\
0 & 3 & 9 & \\
0 & 0 & 3 & \\
\end{bmatrix}
,~
z:
\begin{bmatrix}
0 & 0 & 0 & \\
0 & 3 & 0 & \\
0 & 9 & 3 & \\
0 & 0 & 3 & \\
\end{bmatrix}
,~
1:
\begin{bmatrix}
0 & 0 & 0 & \\
1 & 0 & 0 & \\
0 & 4 & 0 & \\
0 & 0 & 1 & \\
\end{bmatrix}
$$
$$
F_4=x^{4} y^{4} z^{4} + 4 x^{4} y^{3} z^{3} + 6 x^{4} y^{3} z^{2} + 4 x^{4} y^{3} z + x^{4} y^{3} + 11 x^{3} y^{3} z^{4} + 28 x^{3} y^{3} z^{3} + 18 x^{3} y^{3} z^{2} + 4 x^{3} y^{3} z + 16 x^{3} y^{2} z^{3} + 48 x^{3} y^{2} z^{2} + 40 x^{3} y^{2} z + 11 x^{3} y^{2} + 11 x^{2} y^{2} z^{4} + 40 x^{2} y^{2} z^{3} + 48 x^{2} y^{2} z^{2} + 16 x^{2} y^{2} z + 4 x^{2} y z^{3} + 18 x^{2} y z^{2} + 28 x^{2} y z + 11 x^{2} y + x y z^{4} + 4 x y z^{3} + 6 x y z^{2} + 4 x y z + x
$$
$$
z^{4}:
\begin{bmatrix}
0 & 0 & 0 & 0 & \\
1 & 0 & 0 & 0 & \\
0 & 11 & 0 & 0 & \\
0 & 0 & 11 & 0 & \\
0 & 0 & 0 & 1 & \\
\end{bmatrix}
,~
z^{3}:
\begin{bmatrix}
0 & 0 & 0 & 0 & \\
0 & 4 & 0 & 0 & \\
0 & 4 & 40 & 0 & \\
0 & 0 & 16 & 28 & \\
0 & 0 & 0 & 4 & \\
\end{bmatrix}
,~
z^{2}:
\begin{bmatrix}
0 & 0 & 0 & 0 & \\
0 & 6 & 0 & 0 & \\
0 & 18 & 48 & 0 & \\
0 & 0 & 48 & 18 & \\
0 & 0 & 0 & 6 & \\
\end{bmatrix}
,~
z:
\begin{bmatrix}
0 & 0 & 0 & 0 & \\
0 & 4 & 0 & 0 & \\
0 & 28 & 16 & 0 & \\
0 & 0 & 40 & 4 & \\
0 & 0 & 0 & 4 & \\
\end{bmatrix}
,~
1:
\begin{bmatrix}
0 & 0 & 0 & 0 & \\
1 & 0 & 0 & 0 & \\
0 & 11 & 0 & 0 & \\
0 & 0 & 11 & 0 & \\
0 & 0 & 0 & 1 & \\
\end{bmatrix}
$$
$$
F_5=x^{5} y^{5} z^{5} + 5 x^{5} y^{4} z^{4} + 10 x^{5} y^{4} z^{3} + 10 x^{5} y^{4} z^{2} + 5 x^{5} y^{4} z + x^{5} y^{4} + 26 x^{4} y^{4} z^{5} + 75 x^{4} y^{4} z^{4} + 70 x^{4} y^{4} z^{3} + 30 x^{4} y^{4} z^{2} + 5 x^{4} y^{4} z + 55 x^{4} y^{3} z^{4} + 190 x^{4} y^{3} z^{3} + 230 x^{4} y^{3} z^{2} + 125 x^{4} y^{3} z + 26 x^{4} y^{3} + 66 x^{3} y^{3} z^{5} + 275 x^{3} y^{3} z^{4} + 410 x^{3} y^{3} z^{3} + 250 x^{3} y^{3} z^{2} + 55 x^{3} y^{3} z + 55 x^{3} y^{2} z^{4} + 250 x^{3} y^{2} z^{3} + 410 x^{3} y^{2} z^{2} + 275 x^{3} y^{2} z + 66 x^{3} y^{2} + 26 x^{2} y^{2} z^{5} + 125 x^{2} y^{2} z^{4} + 230 x^{2} y^{2} z^{3} + 190 x^{2} y^{2} z^{2} + 55 x^{2} y^{2} z + 5 x^{2} y z^{4} + 30 x^{2} y z^{3} + 70 x^{2} y z^{2} + 75 x^{2} y z + 26 x^{2} y + x y z^{5} + 5 x y z^{4} + 10 x y z^{3} + 10 x y z^{2} + 5 x y z + x
$$
$$
z^{5}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & \\
0 & 1 & 0 & 0 & 0 & 0 & \\
0 & 0 & 26 & 0 & 0 & 0 & \\
0 & 0 & 0 & 66 & 0 & 0 & \\
0 & 0 & 0 & 0 & 26 & 0 & \\
0 & 0 & 0 & 0 & 0 & 1 & \\
\end{bmatrix}
,~
z^{4}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & \\
0 & 5 & 0 & 0 & 0 & \\
0 & 5 & 125 & 0 & 0 & \\
0 & 0 & 55 & 275 & 0 & \\
0 & 0 & 0 & 55 & 75 & \\
0 & 0 & 0 & 0 & 5 & \\
\end{bmatrix}
,~
z^{3}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & \\
0 & 10 & 0 & 0 & 0 & \\
0 & 30 & 230 & 0 & 0 & \\
0 & 0 & 250 & 410 & 0 & \\
0 & 0 & 0 & 190 & 70 & \\
0 & 0 & 0 & 0 & 10 & \\
\end{bmatrix}
$$
$$
z^{2}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & \\
0 & 10 & 0 & 0 & 0 & \\
0 & 70 & 190 & 0 & 0 & \\
0 & 0 & 410 & 250 & 0 & \\
0 & 0 & 0 & 230 & 30 & \\
0 & 0 & 0 & 0 & 10 & \\
\end{bmatrix}
,~
z:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & \\
0 & 5 & 0 & 0 & 0 & \\
0 & 75 & 55 & 0 & 0 & \\
0 & 0 & 275 & 55 & 0 & \\
0 & 0 & 0 & 125 & 5 & \\
0 & 0 & 0 & 0 & 5 & \\
\end{bmatrix}
,~
1:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & \\
1 & 0 & 0 & 0 & 0 & \\
0 & 26 & 0 & 0 & 0 & \\
0 & 0 & 66 & 0 & 0 & \\
0 & 0 & 0 & 26 & 0 & \\
0 & 0 & 0 & 0 & 1 & \\
\end{bmatrix}
$$
$$
F_6=x^{6} y^{6} z^{6} + 6 x^{6} y^{5} z^{5} + 15 x^{6} y^{5} z^{4} + 20 x^{6} y^{5} z^{3} + 15 x^{6} y^{5} z^{2} + 6 x^{6} y^{5} z + x^{6} y^{5} + 57 x^{5} y^{5} z^{6} + 186 x^{5} y^{5} z^{5} + 225 x^{5} y^{5} z^{4} + 140 x^{5} y^{5} z^{3} + 45 x^{5} y^{5} z^{2} + 6 x^{5} y^{5} z + 156 x^{5} y^{4} z^{5} + 630 x^{5} y^{4} z^{4} + 1000 x^{5} y^{4} z^{3} + 810 x^{5} y^{4} z^{2} + 336 x^{5} y^{4} z + 57 x^{5} y^{4} + 302 x^{4} y^{4} z^{6} + 1416 x^{4} y^{4} z^{5} + 2550 x^{4} y^{4} z^{4} + 2200 x^{4} y^{4} z^{3} + 930 x^{4} y^{4} z^{2} + 156 x^{4} y^{4} z + 396 x^{4} y^{3} z^{5} + 1980 x^{4} y^{3} z^{4} + 3840 x^{4} y^{3} z^{3} + 3600 x^{4} y^{3} z^{2} + 1656 x^{4} y^{3} z + 302 x^{4} y^{3} + 302 x^{3} y^{3} z^{6} + 1656 x^{3} y^{3} z^{5} + 3600 x^{3} y^{3} z^{4} + 3840 x^{3} y^{3} z^{3} + 1980 x^{3} y^{3} z^{2} + 396 x^{3} y^{3} z + 156 x^{3} y^{2} z^{5} + 930 x^{3} y^{2} z^{4} + 2200 x^{3} y^{2} z^{3} + 2550 x^{3} y^{2} z^{2} + 1416 x^{3} y^{2} z + 302 x^{3} y^{2} + 57 x^{2} y^{2} z^{6} + 336 x^{2} y^{2} z^{5} + 810 x^{2} y^{2} z^{4} + 1000 x^{2} y^{2} z^{3} + 630 x^{2} y^{2} z^{2} + 156 x^{2} y^{2} z + 6 x^{2} y z^{5} + 45 x^{2} y z^{4} + 140 x^{2} y z^{3} + 225 x^{2} y z^{2} + 186 x^{2} y z + 57 x^{2} y + x y z^{6} + 6 x y z^{5} + 15 x y z^{4} + 20 x y z^{3} + 15 x y z^{2} + 6 x y z + x
$$
$$
z^{6}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & \\
0 & 0 & 57 & 0 & 0 & 0 & 0 & \\
0 & 0 & 0 & 302 & 0 & 0 & 0 & \\
0 & 0 & 0 & 0 & 302 & 0 & 0 & \\
0 & 0 & 0 & 0 & 0 & 57 & 0 & \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & \\
\end{bmatrix}
,~
z^{5}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & \\
0 & 6 & 0 & 0 & 0 & 0 & \\
0 & 6 & 336 & 0 & 0 & 0 & \\
0 & 0 & 156 & 1656 & 0 & 0 & \\
0 & 0 & 0 & 396 & 1416 & 0 & \\
0 & 0 & 0 & 0 & 156 & 186 & \\
0 & 0 & 0 & 0 & 0 & 6 & \\
\end{bmatrix}
,~
z^{4}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & \\
0 & 15 & 0 & 0 & 0 & 0 & \\
0 & 45 & 810 & 0 & 0 & 0 & \\
0 & 0 & 930 & 3600 & 0 & 0 & \\
0 & 0 & 0 & 1980 & 2550 & 0 & \\
0 & 0 & 0 & 0 & 630 & 225 & \\
0 & 0 & 0 & 0 & 0 & 15 & \\
\end{bmatrix}
,~
z^{3}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & \\
0 & 20 & 0 & 0 & 0 & 0 & \\
0 & 140 & 1000 & 0 & 0 & 0 & \\
0 & 0 & 2200 & 3840 & 0 & 0 & \\
0 & 0 & 0 & 3840 & 2200 & 0 & \\
0 & 0 & 0 & 0 & 1000 & 140 & \\
0 & 0 & 0 & 0 & 0 & 20 & \\
\end{bmatrix}
$$
$$
z^{2}:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & \\
0 & 15 & 0 & 0 & 0 & 0 & \\
0 & 225 & 630 & 0 & 0 & 0 & \\
0 & 0 & 2550 & 1980 & 0 & 0 & \\
0 & 0 & 0 & 3600 & 930 & 0 & \\
0 & 0 & 0 & 0 & 810 & 45 & \\
0 & 0 & 0 & 0 & 0 & 15 & \\
\end{bmatrix}
,~
z:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & \\
0 & 6 & 0 & 0 & 0 & 0 & \\
0 & 186 & 156 & 0 & 0 & 0 & \\
0 & 0 & 1416 & 396 & 0 & 0 & \\
0 & 0 & 0 & 1656 & 156 & 0 & \\
0 & 0 & 0 & 0 & 336 & 6 & \\
0 & 0 & 0 & 0 & 0 & 6 & \\
\end{bmatrix}
,~
1:
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & \\
1 & 0 & 0 & 0 & 0 & 0 & \\
0 & 57 & 0 & 0 & 0 & 0 & \\
0 & 0 & 302 & 0 & 0 & 0 & \\
0 & 0 & 0 & 302 & 0 & 0 & \\
0 & 0 & 0 & 0 & 57 & 0 & \\
0 & 0 & 0 & 0 & 0 & 1 & \\
\end{bmatrix}
$$