A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.
1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 6, 8, 0, 0, 10, 24, 30, 40, 0, 0, 26, 160, 144, 180, 160, 0, 0, 76, 1140, 1120, 1008, 840, 700, 0, 0, 232, 8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764, 80864, 80892, 82080, 53760, 30240, 19656, 12768, 0, 0, 2620, 809856, 808640, 808920, 547200, 336000, 157248, 95760, 55680, 0, 0, 9496
Offset: 0
Examples
Triangle T(n,k) begins:
1;
0, 1;
0, 0, 2;
2, 0, 0, 4;
6, 8, 0, 0, 10;
24, 30, 40, 0, 0, 26;
160, 144, 180, 160, 0, 0, 76;
1140, 1120, 1008, 840, 700, 0, 0, 232;
8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764;
...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
-
Maple
b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)* binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n))) end: T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0): seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Oct 28 2024 -
Mathematica
b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[b[n-j, t]* Binomial[n-1, j-1]*(j-1)!, {j, If[t == 1, Range @ Min[2, n], Range[3, n]]}]]; T[n_, k_] := Binomial[n, k]*b[k, 1]*b[n-k, 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)