A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 4, 6, 0, 2, 16, 20, 24, 0, 14, 44, 80, 108, 120, 0, 90, 200, 384, 544, 672, 720, 0, 646, 1288, 2240, 3264, 4128, 4800, 5040, 0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320, 0, 47622, 78652, 123456, 176832, 231936, 280512, 323520, 352800, 362880

Offset

0,6

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = n! for k>=n.

Links

Alois P. Heinz, Rows n = 0..20, flattened

Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.

Wikipedia, Permutation

Example

Triangle T(n,k) begins:
  1;
  0,    1;
  0,    0,    2;
  0,    0,    4,     6;
  0,    2,   16,    20,    24;
  0,   14,   44,    80,   108,   120;
  0,   90,  200,   384,   544,   672,   720;
  0,  646, 1288,  2240,  3264,  4128,  4800,  5040;
  0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320;
  ...

Crossrefs

Columns k=0-10 (for n>=k) give: A000007, A002464, A110128, A117574, A189255, A189256, A189271, A360384, A360386, A360462, A360463.

Main diagonal gives A000142.

T(2n,n) gives A189849.

T(n+1,n) gives 4*A138772(n).

T(n+2,n) gives 16*A333804(n).

Cf. A000170 (condition is satisfied for all k), A248686 (p(i) at distance k are sorted).

Sequence in context: A244138 A284611 A282551 * A056676 A352450 A098699

Adjacent sequences: A333703 A333704 A333705 * A333707 A333708 A333709

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 02 2020

Status

approved