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

0,9

Comments

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

T(0,0) = 1 by convention.

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

Formula

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

Example

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;
  ...

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<n do ok, i, k:=
             not i in s xor k in s, i-1, k+1 od;
           r:= r+ `if`(ok, b(s minus {j}), 0)
         od; r
      fi
    end:
T:= (n, 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 *)

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.

Sequence in context: A127597 A167749 A104770 * A110280 A061009 A144106

Adjacent sequences: A296526 A296527 A296528 * A296530 A296531 A296532

Keyword

nonn,tabl,look,changed

Author

Alois P. Heinz, Dec 14 2017

Status

approved