A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 6, 5, 1, 1, 1, 2, 6, 14, 8, 1, 1, 1, 2, 6, 24, 31, 13, 1, 1, 1, 2, 6, 24, 78, 73, 21, 1, 1, 1, 2, 6, 24, 120, 230, 172, 34, 1, 1, 1, 2, 6, 24, 120, 504, 675, 400, 55, 1, 1, 1, 2, 6, 24, 120, 720, 1902, 2069, 932, 89, 1, 1, 1, 2, 6, 24, 120, 720, 3720, 6902, 6404, 2177, 144, 1

Offset

0,9

Comments

A(n,k) counts permutations p of [n] such that |p(j)-j| <= k for all j in [n].

Links

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

Torleiv Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104.

Formula

A(n,k) = Sum_{j=0..k} A130152(n,j) for n > 0, A(0,k) = 1.

Example

A(4,1) = 5: 1234, 1243, 1324, 2134, 2143.
A(5,2) = 31: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14253, 14325, 14523, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24135, 24153, 31245, 31254, 31425, 31524, 32145, 32154, 34125.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  2,   2,    2,    2,     2,     2,     2,     2, ...
  1,  3,   6,    6,    6,     6,     6,     6,     6, ...
  1,  5,  14,   24,   24,    24,    24,    24,    24, ...
  1,  8,  31,   78,  120,   120,   120,   120,   120, ...
  1, 13,  73,  230,  504,   720,   720,   720,   720, ...
  1, 21, 172,  675, 1902,  3720,  5040,  5040,  5040, ...
  1, 34, 400, 2069, 6902, 17304, 30960, 40320, 40320, ...

Mathematica

A[0, _] = 1;
A[n_ /; n > 0, k_] := A[n, k] = Permanent[Table[If[Abs[i - j] <= k, 1, 0], {i, 1, n}, {j, 1, n}]];
Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1 }] // Flatten (* Jean-François Alcover, Oct 18 2021, after Alois P. Heinz in A130152 *)

Crossrefs

Columns k=0..10 give: A000012, A000045(n+1), A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659.

Rows n=1-2 give: A000012, A040000.

Main diagonal gives A000142.

A(2n,n) gives A048163(n+1).

A(2n+1,n) gives A092552(n+1).

A(n,floor(n/2)) gives A306267.

A(n+2,n) gives A001564.

Cf. A130152.

Sequence in context: A096670 A130461 A225631 * A267482 A130777 A187660

Adjacent sequences: A306206 A306207 A306208 * A306210 A306211 A306212

Keyword

nonn,tabl

Author

Alois P. Heinz, Jan 29 2019

Status

approved