A296529 Number T(n,k) of non-averaging permutations of [n] with first element k; triangle T(n,k), n >= 0, k = 0..n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 3, 3, 2, 0, 2, 5, 6, 5, 2, 0, 5, 6, 13, 13, 6, 5, 0, 10, 10, 16, 32, 16, 10, 10, 0, 28, 26, 36, 51, 51, 36, 26, 28, 0, 24, 50, 62, 74, 76, 74, 62, 50, 24, 0, 50, 50, 134, 138, 161, 161, 138, 134, 50, 50, 0, 124, 120, 146, 302, 345, 386, 345, 302, 146, 120, 124

Offset: 0

Author

Alois P. Heinz, Dec 14 2017

Keywords

• nonn
• tabl
• look

Comments

A non-averaging permutation avoids any 3-term arithmetic progression.

T(0,0) = 1 by convention.

Examples

			T(5,1) = 2: 15324, 15342.
T(5,2) = 5: 21453, 24153, 24315, 24351, 24513.
T(5,3) = 6: 31254, 31524, 31542, 35124, 35142, 35412.
T(5,4) = 5: 42153, 42315, 42351, 42513, 45213.
T(5,5) = 2: 51324, 51342.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  1,  2,   1;
  0,  2,  3,   3,   2;
  0,  2,  5,   6,   5,   2;
  0,  5,  6,  13,  13,   6,   5;
  0, 10, 10,  16,  32,  16,  10,  10;
  0, 28, 26,  36,  51,  51,  36,  26,  28;
  0, 24, 50,  62,  74,  76,  74,  62,  50, 24;
  0, 50, 50, 134, 138, 161, 161, 138, 134, 50, 50;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..99, flattened
• Eric Weisstein's World of Mathematics, Nonaveraging Sequence
• Wikipedia, Arithmetic progression
• Index entries related to non-averaging sequences

Crossrefs

Columns k=0-1 give: A000007, A296530 (for n>0).

Row sums give A003407.

T(n,n) gives A296530.

T(n,ceiling(n/2)) gives A296531.

Cf. A292523.

Programs

Maple

b:= proc(s) option remember; local n, r, ok, i, j, k;
      if nops(s) = 1 then 1
    else n, r:= max(s), 0;
         for j in s minus {n} do ok, i, k:= true, j-1, j+1;
           while ok and i>=0 and k `if`(k=0, 0^n, b({$0..n} minus {k-1})):
seq(seq(T(n, k), k=0..n), n=0..14);

Mathematica

b[s_List] := b[s] = Module[{n = Max[s], r = 0, ok, i, j, k}, If[Length[s] == 1, 1, Do[{ok, i, k} = {True, j-1, j+1}; While[ok && i >= 0 && k < n, {ok, i, k} = {FreeQ[s, i] ~Xor~ MemberQ[s, k], i-1, k+1}]; r = r + If[ok, b[s ~Complement~ {j}], 0], {j, s ~Complement~ {n}}]; r]];
T[0, 0]=1; T[n_, k_] := If[k==0, 0^n, b[Range[0, n] ~Complement~ {k-1}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2017, after Alois P. Heinz *)

Formula

T(n,k) = T(n,n+1-k) > 0 for k=1..n.