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

Author

Alois P. Heinz, Apr 02 2020

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

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

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).