A306506
Number T(n,k) of permutations p of [n] having at least one index i with |p(i)-i| = k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
Original entry on oeis.org
1, 1, 1, 4, 4, 3, 15, 19, 15, 10, 76, 99, 86, 67, 42, 455, 603, 544, 455, 358, 216, 3186, 4248, 3934, 3486, 2921, 2250, 1320, 25487, 34115, 32079, 29296, 25487, 21514, 16296, 9360, 229384, 307875, 292509, 272064, 245806, 214551, 179058, 133800, 75600
Offset: 1
The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have absolute displacement sets {|p(i)-i|, i=1..3}: {0}, {0,1}, {0,1}, {1,2}, {1,2}, {0,2}, respectively. Number 0 occurs four times, 1 occurs four times, and 2 occurs thrice. So row n=3 is [4, 4, 3].
Triangle T(n,k) begins:
1;
1, 1;
4, 4, 3;
15, 19, 15, 10;
76, 99, 86, 67, 42;
455, 603, 544, 455, 358, 216;
3186, 4248, 3934, 3486, 2921, 2250, 1320;
25487, 34115, 32079, 29296, 25487, 21514, 16296, 9360;
...
T(n+2,n+1) gives
A007680 (for n>=0).
-
b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
add(b(s minus {i}, d union {abs(n-i)}), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b({$1..n}, {})):
seq(T(n), n=1..9);
# second Maple program:
T:= proc(n, k) option remember; n!-LinearAlgebra[Permanent](
Matrix(n, (i, j)-> `if`(abs(i-j)=k, 0, 1)))
end:
seq(seq(T(n, k), k=0..n-1), n=1..9);
-
T[n_, k_] := n!-Permanent[Table[If[Abs[i-j]==k, 0, 1], {i, 1, n}, {j, 1, n} ]];
Table[T[n, k], {n, 1, 9}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, May 01 2019, from 2nd Maple program *)
A306512
Number A(n,k) of permutations p of [n] having no index i with |p(i)-i| = k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 9, 1, 1, 2, 3, 5, 44, 1, 1, 2, 6, 9, 21, 265, 1, 1, 2, 6, 14, 34, 117, 1854, 1, 1, 2, 6, 24, 53, 176, 792, 14833, 1, 1, 2, 6, 24, 78, 265, 1106, 6205, 133496, 1, 1, 2, 6, 24, 120, 362, 1554, 8241, 55005, 1334961
Offset: 0
A(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
A(4,1) = 5: 1234, 1432, 3214, 3412, 4231.
A(4,2) = 9: 1234, 1243, 1324, 2134, 2143, 2341, 4123, 4231, 4321.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 2, 2, 2, 2, 2, ...
2, 2, 3, 6, 6, 6, 6, 6, ...
9, 5, 9, 14, 24, 24, 24, 24, ...
44, 21, 34, 53, 78, 120, 120, 120, ...
265, 117, 176, 265, 362, 504, 720, 720, ...
1854, 792, 1106, 1554, 2119, 2790, 3720, 5040, ...
-
A:= proc(n, k) option remember; `if`(k>=n, n!, LinearAlgebra[
Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)=k, 0, 1))))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(s, k) option remember; (n-> `if`(n=0, 1, add(
`if`(abs(i-n)=k, 0, b(s minus {i}, k)), i=s)))(nops(s))
end:
A:= (n, k)-> `if`(k>=n, n!, b({$1..n}, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
A[n_, k_] := If[k > n, n!, Permanent[Table[If[Abs[i-j] == k, 0, 1], {i, 1, n}, {j, 1, n}]]]; A[0, 0] = 1;
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 05 2021, from first Maple program *)
b[s_, k_] := b[s, k] = With[{n = Length[s]}, If[n == 0, 1, Sum[
If[Abs[i-n] == k, 0, b[s ~Complement~ {i}, k]], {i, s}]]];
A[n_, k_] := If[k >= n, n!, b[Range@n, k]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Sep 01 2021, from second Maple program *)
A306675
Number of permutations p of [2n] having at least one index i with |p(i)-i| = n.
Original entry on oeis.org
0, 1, 15, 455, 25487, 2293839, 302786759, 55107190151, 13225725636255, 4047072044694047, 1537887376983737879, 710503968166486900119, 392198190427900768865711, 254928823778135499762712175, 192726190776270437820610404327, 167671785975355280903931051764519
Offset: 0
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b:= proc(n, k) b(n, k):= `if`(k=0, n!, b(n+1, k-1) -b(n, k-1)) end:
a:= n-> (2*n)! -b(0, 2*n):
seq(a(n), n=0..16);
-
b[n_, k_] := b[n, k] = If[k == 0, n!, b[n + 1, k - 1] - b[n, k - 1]];
a[n_] := (2n)! - b[0, 2n];
a /@ Range[0, 16] (* Jean-François Alcover, Apr 02 2021, after Alois P. Heinz *)
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