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A306543
Number T(n,k) of permutations p of [n] such that |p(j)-j| >= k (for all j in [n]); triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.
2
1, 1, 2, 1, 6, 2, 24, 9, 1, 120, 44, 4, 720, 265, 29, 1, 5040, 1854, 206, 8, 40320, 14833, 1708, 112, 1, 362880, 133496, 15702, 1168, 16, 3628800, 1334961, 159737, 13365, 436, 1, 39916800, 14684570, 1780696, 159414, 6984, 32, 479001600, 176214841, 21599745, 2036488, 114124, 1708, 1
OFFSET
0,3
LINKS
Wikipedia, Permutation
FORMULA
T(n,k) = Sum_{j=k..floor(n/2)} A299789(n,j) for n > 0.
EXAMPLE
Triangle T(n,k) begins:
1;
1;
2, 1;
6, 2;
24, 9, 1;
120, 44, 4;
720, 265, 29, 1;
5040, 1854, 206, 8;
40320, 14833, 1708, 112, 1;
362880, 133496, 15702, 1168, 16;
3628800, 1334961, 159737, 13365, 436, 1;
39916800, 14684570, 1780696, 159414, 6984, 32;
479001600, 176214841, 21599745, 2036488, 114124, 1708, 1;
...
MAPLE
T:= proc(n, k) option remember; `if`(n=0, 1, LinearAlgebra[
Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))
end:
seq(seq(T(n, k), k=0..floor(n/2)), n=0..12);
MATHEMATICA
T[n_, k_] := T[n, k] = If[n==0, 1, Permanent[Table[
If[Abs[i-j] >= k, 1, 0], {i, n}, {j, n}]]];
Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 26 2021, after Alois P. Heinz *)
CROSSREFS
Columns k=0-6 give (offsets may differ): A000142, A000166, A001883, A075851, A075852, A183242, A183243.
T(2n,n) gives A000012.
T(2n+1,n) gives A000079.
T(2n+2,n) gives A183245 for n > 0.
T(2n+3,n) gives A183246 for n > 0.
T(2n+4,n) gives A183247 for n > 0.
Sequence in context: A325703 A321898 A284434 * A243484 A181811 A284431
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Feb 22 2019
STATUS
approved